computationally efficient predictive direct torque...
TRANSCRIPT
Computationally Efficient Predictive Direct Torque Control S trategyfor PMSGs Without Weighting FactorsMohamed Abdelrahem a, Hisham Eldeeb b, Christoph Hackl c, Ralph Kennel a, and Jose Rodriguez d
aInstitute for Electrical Drive Systems and Power Electronics, Technical University of Munich (TUM), Germany,E-mail: [email protected] (corresponding author), [email protected] Group “Control of Renewable Energy Systems (CRES)”, Munich School of Engineering, TUM, Germany,E-mail: [email protected] of Electrical Engineering and Information Technology, Munich University of Applied Sciences,Germany, E-mail: [email protected] of Engineering, Universidad Andres Bello, Santiago, Chile, E-mail: [email protected]
Abstract
This paper proposes a computationally effi-cient Predictive Torque Control (PTC) techniquefor permanent-magnet synchronous generators(PMSGs) without weighting factors. The proposedcontrol strategy is based on computing the q-axisreference current from the demanded torque. Fur-thermore, the d-axis reference current is set to zeroto achieve the maximum torque per ampere (MTPA)operation of the PMSG. Then, the reference voltagevector (VV) is directly computed from the referencecurrent vector using the deadbeat principle. Finally,according to the location of this reference VV, onlythree evaluations of the cost function are required.The cost function includes only the error betweenthe reference VV and the candidates ones, whicheliminates the need of weighting factors. Therefore,the proposed control scheme overcomes the follow-ing drawbacks of the classical PTC: 1) High calcu-lation burden, and 2) tuning of the weighting factors.Experimental results using a dSPACE DS1007 real-time platform and a 14.5 kW PMSG are presented toverify the feasibility of the proposed control method.
1. Introduction
Recently, the use of renewable energy generationunits are increasing. In particular, wind power isconsidered one of the most promising technolo-gies for electrical power generation. Variable-speedwind generators can be divided into: (i) doubly-fedinduction generators (DFIGs), and (ii) permanent-magnet synchronous generators (PMSGs). Themain feature of DFIGs is the utilization of a partial-scale (approx. 30% of the machine rating) back-to-back (B2B) power converter to tie the rotor wind-
ings of the DFIG with the grid [1]-[5]. However, aheavy multi-stage gear box, which requires mainte-nance and reduces the system reliability, is essen-tial. Furthermore, the sensitivity to faults/voltagedips on the grid side is another drawback of theDFIGs [6], [7]. Therefore, the direct-driven PMSGwith a full-scale B2B power converter is a viable al-ternative for variable-speed wind turbine technolo-gies [1], [8]-[9].
The commonly adopted control schemes forPMSGs include vector control and direct torquecontrol (DTC). Compared with the classical vec-tor control techniques, DTC owns several featureslike elimination of coordinate transformation, ro-bustness to parameter variations, and quick tran-sient performance [10]. However, DTC schemeswith hysteresis comparators suffer from the follow-ing disadvantages: large torque ripple and highsampling requirements for digital implementation.
Recently, model predictive control (MPC) strate-gies have been spread out across various fieldsincluding power electronics, electrical drives, andvariable-speed wind generators [11], [12]. Predic-tive deadbeat (DB)-DTC techniques have been ap-plied for permanent-magnet synchronous machines(PMSMs) in [13]. However, the main disadvantageof the DB-DTC strategies is their sensitivity to vari-ations of the machine parameters. Another alterna-tive is the continuous-control-set MPC (CCS-MPC),which considers the model of the system to pre-dict its future behavior over a given prediction hori-zon. Then, the voltage vector that minimizes a cer-tain cost function is selected. Finally, a modulationstage is used to generate the switching signals ofthe converter. The CCS-MPC has been utilizedto control the PMSMs in [14], [15]. However, its
high computational load is the main drawback. Tak-ing into account the discrete nature of the powerconverters/inverters, the so called finite-control-setMPC (FCS-MPC) has been proposed. This tech-nique (i.e. FCS-MPC) has been applied for vari-ous applications like voltage source converters andmotor drive systems due to its advantages such asfast dynamic response, simple implementation, andhandling of nonlinearities and constraints [11], [12].Predictive current control, which belongs to FCS-MPC, has been applied for PMSGs in [9], [16]-[19]and Predictive Torque Control (PTC) has also beenemployed to control the PMSMs in [20]-[26].
Generally, in PTC schemes, the control variablesare the torque and stator flux [20], [21]. However,prediction of the stator flux in the next sampling in-stant and calculation of the reference stator flux ac-cording to the maximum torque per ampere (MTPA)trajectory increase the complexity of the control sys-tem. Furthermore, a weighting factor is employedto penalize the stator flux magnitude error. Thisweighting factor has a significant impact on the con-trol response, particularly on the harmonic currentdistortions. Furthermore, tuning of this weightingfactor is normally realized by trail-and-error method,which is a time consuming technique. An empiricalprocedure to calculate the suitable weighting fac-tors is proposed in [22]. However, this procedurelacks sufficient theoretical support. In [23], the prin-ciple of torque ripple minimization is used to com-pute the required weighting factor online. A fuzzydecision making strategy is proposed in [24] and amulti-objective ranking based method is presentedin [25]. However, the main drawback of those meth-ods is the required high calculation burden. To re-duce the calculation load, the computation of theweighting factor is realized by a simple look up tabletechnique in [26]. However, this method requiressubstantial offline calculation.
In this paper, firstly, the d-axis current of the PMSGis selected as the second control variable besidethe torque. Accordingly, the MTPA operation is re-alized easily by setting the reference current of thed-axis to zero, which slightly reduces the calculationload. Secondly, in order to avoid using weightingfactors in the cost function, the reference current ofthe q-axis is determined according to the demandedtorque. Furthermore, the reference voltage vector
dcu
]1[, +kidrefs
]1[ +kids
asi
Wind Turbine
CostFunction
]1[ +kTe
][kids][kiq
s
abcbsi c
si
PMSG
dq
abcsPrediction
Model
]1[* +kTe
7
7
][kudqs
sR sL pmψ
7][krω
pn
][krφ
dt
d][krω
Fig. 1: Traditional PTC strategy for surface-mountedPMSGs.
(VV) is directly computed from the reference d- andq-axis currents using a deadbeat function. Thirdly,in order to reduce the computational effort, the sec-tor where the reference VV is located is determined.Therefore, three evaluations of the cost function areonly required to find the optimal VV to apply in thenext sampling interval. The cost function containsthe error between the reference VV and the can-didates VVs. Accordingly, no weighting factors arerequired. The performance of the proposed con-trol technique is validated experimentally and its re-sponse is compared with that of the conventionalPTC scheme.
2. Modeling of the PMSG
In direct-drive variable speed wind turbine systems(WTSs), the PMSG is mechanically coupled to thewind turbine via a stiff shaft (see Fig. 1). The sta-tor windings of the PMSG are tied via a B2B powerconverter and a filter to the grid. The machine-sideconverter (MSC) is utilized to control the electro-magnetic torque of the PMSG and to achieve theMTPA criteria. The continuous-time model of athree-phase surface-mounted PMSG can be writtenin the synchronously rotating dq-reference frame asfollows [27]:
uds(t) = Rsids(t) +
ddtψds (t)− ωrψ
qs(t),
uqs(t) = Rsiqs(t) +
ddtψqs(t) + ωrψ
ds (t),
(1)
where uds , uqs, ids , iqs, ψds , ψq
s are the d- and q-axescomponents of the stator voltage (in V), current (inA), and flux (in Wb) of the PMSG, respectively. Rs
is the stator resistance (in Ω) of the PMSG and ωr =
npωm is the electrical angular speed of the rotor (inrad/s), where np is the pole pair number and ωm isthe mechanical angular speed of the rotor.
The stator flux linkage of the PMSG can be writtenas follows
ψds (t) = Lsi
ds(t) + ψpm & ψq
s(t) = Lsiqs(t). (2)
In (2), Ls is the stator inductance (in H) of the PMSGand ψpm is the permanent-magnet flux linkage (inWb). The dynamics of the mechanics of the (stiff)wind turbine system are given by
ddtωm(t) = 1
Θ(Te(t)− Tm(t)− νωm(t)),Te(t) =
32npψpmi
qs(t).
(3)
In (3), Te is the electro-magnetic torque (in Nm) andTm is the mechanical torque produced by the windturbine. Θ is the overall rotor inertia (in kg/m2) ofthe wind turbine and PMSG, and ν is the viscousfriction coefficient (in Nms; see [28, Sec. 11.1.5]).
3. Traditional PTC scheme
The structure of the traditional PTC scheme is illus-trated in Fig. 1. To design this control technique, (2)is inserted into (1) and solved for d
dtidqs giving
ddtids(t) = −Rs
Lsids(t) + ωri
qs(t) +
1Lsuds(t),
ddtiqs(t) = −Rs
Lsiqs(t)− ωri
ds(t)−
ωr
Lsψpm + 1
Lsuqs(t).
(4)For predicting the currents at a future sampling in-terval, a discrete-time model is required. Thus, theforward Euler method with a sampling time Ts ≪1 s is applied to the time-continuous model in (4).Hence, the discrete-time model of the PMSG in therotating dq-reference frame can be written as fol-lows [27]
ids [k + 1] = (1− TsRs
Ls)ids [k] + ωrTsi
qs[k] +
Ts
Lsuds [k],
iqs[k + 1] = (1− TsRs
Ls)iqs[k]− ωrTsi
ds [k]−
ωrTs
Lsψpm
+ Ts
Lsuqs[k].
(5)The electro-magnetic torque can be predicted by
Te[k + 1] =3
2npψpmi
qs[k + 1]. (6)
In this work, the control variable are the torque andthe d-axis current, as shown in Fig. 1. The sta-tor voltage u
dqs [k] of the PMSG can be expressed
as a function of the switching state vector sabc[k] ∈
uφ
6
αβrefsu ,
011
110010
001 101
000111 αβ1,su
αβ2,suαβ
3,su
αβ4,su
αβ5,su αβ
6,su
α
β
100
5
4
3
2
1αβ
0,su
Fig. 2: Different switching combinations of 2-level volt-age source converter.
0, 13 of the power converter as follows [28, Chap-ter 14]:
udqs [k] =
[cos(φr) sin(φr)− sin(φr) cos(φr)
]
︸ ︷︷ ︸
=:TP (φr)−1
2
3
[1 − 1
2 − 12
0√32 −
√32
]
︸ ︷︷ ︸
=:TC
1
3udc[k]
2 −1 −1−1 2 −1−1 −1 2
sabc[k]
︸ ︷︷ ︸
=:uabcs
[k]
,
(7)where T P (φr)
−1 and TC are the Park and Clarketransformation matrices, respectively. udc is the DC-link voltage (in V) and u
abcs = (uas , u
bs, u
cs)
⊤ is thestator phase voltage vector (in V) applied to thePMSG in the abc-reference frame. φr = npφm is theelectrical rotor position of the PMSG (in rad). Con-sidering all the possible combinations of the switch-ing state vector sabc as shown in Fig. 2, seven differ-ent voltage vectors can be obtained. Those sevenvoltage vectors can be used to predict seven fu-ture values of the current ids [k + 1] and the electro-magnetic torque Te[k + 1]. Then, the following costfunction
g =∣∣T ∗
e [k + 1]− Te[k + 1]∣∣+ γ
∣∣ids,ref [k + 1]− ids [k + 1]
∣∣
+
0 if Te[k + 1] ≤ Te,max[k + 1],
∞ if Te[k + 1] > Te,max[k + 1],
+
0 if√
ids [k + 1]2 + iqs[k + 1]2 ≤ is,max,
∞ if√
ids [k + 1]2 + ids [k + 1]2 > is,max,
(8)with soft constraints is employed to select the opti-
mal switching state vector which minimizes the costfunction. This optimal switching vector is then ap-plied at the next sampling instant. In (8), γ is aweighting factor, Te,max is the maximum allowabletorque of the PMSG, and is,max in the maximumcurrent of the stator.
dcu
asi
Wind Turbine
CostFunction
dsi
qsi
abcbsi c
si
PMSG
dq
abcsReference VV
calculation and sector
identification
*eT
sR sL pmψ
pnrφ
dt
d
rω
pmpnψ32
drefsi ,
qrefsi ,
αβrefsu ,
Sector
Fig. 3: Proposed PTC strategy without weighting factorsfor surface-mounted PMSGs.
The traditional PTC strategy suffers from the fol-lowing disadvantages: 1) Tuning of the weightingfactor γ, which is normally tuned by trial-and-errormethod, and 2) high calculation load.
4. Proposed PTC strategy
The proposed PTC is illustrated in Fig. 3. Firstly, theq-axis reference current iqs,ref [k + 1] can by com-puted directly from the reference electro-magnetictorque T ∗
e [k + 1] as follows
iqs,ref [k + 1] =2T ∗
e [k + 1]
3npψpm
. (9)
Secondly, using the reference current idqs,ref [k + 1],
the reference VV udqs,ref [k] can be directly calculated
using the deadbeat principle as follows
uds,ref [k] = Rsids [k] + Ls
ids,ref
[k+1]−ids [k]
Ts
−ωr[k]Lsiqs[k],
uqs,ref [k] = Rsiqs[k] + Ls
iqs,ref
[k+1]−iqs[k]
Ts
+ωr[k]Lsids [k] + ωr[k]ψpm.
(10)
The magnitude
us[k]=‖udqs,ref [k]‖=
√
uds,ref [k]2 + uqs,ref [k]
2
of the reference voltage vector udqs,ref [k] is calcu-
lated and compared with the maximally availableoutput voltage magnitude us,max of the voltagesource converter which depends on the dc-link volt-age udc. If the magnitude is greater than this value,the reference voltages should be adjusted as fol-lows
udqs,ref [k] =
udqs,ref [k], us[k] ≤ us,max
us,max
us[k]udqs,ref [k], us[k] > us,max.
(11)
Fig. 4: Laboratory setup to validate the proposed PTCscheme for PMSGs.
Tab. 1: PMSG parameters.
Name Symbol ValueRated power prated 14.5 kWStator line-line voltage us,rated 400V
DC-link voltage udc 560V
Mechanical speed ωm,rated 209 rad/sStator resistance Rs 0.15ΩStator inductance Ls 3.4mH
PM flux linkage ψpm 0.3753Wb
Pole pairs np 3
This reference VV udqs,ref [k] is transformed to the
stationary reference frame αβ using the Park trans-formation. Therefore, its location can be identi-fied as shown in Fig. 2 using its angle φu[k] =
atan2(uβs,ref [k], uαs,ref [k]). The new cost function
has the form
gnew =∣∣uαs,ref [k]− uαs [k]
∣∣+
∣∣uβs,ref [k]− uβs [k]
∣∣. (12)
Based on the location of the reference VV uαβs,ref [k],the six sectors are defined, which are illustrated inFig. 2. For clarification, when φu[k] ∈ [0, π3 ], thenthe reference VV is located in sector 1 and the onlyreasonable candidate VVs are uαβs,0, uαβs,1, and uαβs,2.Hence, (12) is evaluated for only three times to ob-tain the optimal VV. Moreover, there is no need touse a weighting factor in the cost function. Accord-ingly, the proposed PTC overcomes the disadvan-tages of the traditional one.
5. Description of Laboratory Setup
The proposed and traditional PTC techniques havebeen experimentally implemented. The setup con-sists of a 14.5 kW PMSG driven by a two-levelvoltage source converter (VSC). A 9.5 kW reluc-
-30
-20
-10
-20
-10
0
10
20
0 1 2 3 4 5 6 7 8 9-20
-15
-10
-5
Fig. 5: Performance of the traditional PTC at different values of the weighting factor γ.
-60
-30
0
30
-10
0
10
0 0.5 1 1.5 2 2.5 3 3.5 4-40
-20
0
20
-60
-30
0
30
-10
0
10
0 0.5 1 1.5 2 2.5 3 3.5 4-40
-20
0
20
Fig. 6: Experimental results for step changes in the electro-magnetic torque Te: (a) Proposed PTC, and (b) tradi-tional PTC.
tance synchronous machine (RSM) driven by an-other two-level VSC is employed to emulate thevariable-speed wind turbine dynamics and is con-trolled using a nonlinear current PI-based field-oriented control (FOC) technique [29]. The two ma-chines (i.e. PMSG and RSM) are coupled through atorque sensor as illustrated in Fig. 4. The proposedand traditional PTC schemes are implemented ona dSPACE DS1007 real-time system using MAT-LAB/Simulink and Control Desk software. The sam-pling frequency is set to 11 kHz. An incremental en-coder is used to measure the rotor position of thePMSG. Three current sensors and one voltage sen-sor are used to measure the stator currents of the
PMSG and the DC-link voltage, respectively. Theexperimental setup is depicted in Fig. 4 and the pa-rameters of the PMSG are listed in Table 1.
6. Experimental Results
The reference value of the electro-magnetic torqueT ⋆e is selected to be lower than the rated value of the
RSM (i.e. T ratedRSM = 61Nm) and the reference value
of the d-axis current ids,ref is set to zero to achievethe MTPA condition. Fig. 5 illustrates the perfor-mance of the traditional PTC at different values ofthe weighting factor γ. The electro-magnetic torqueis set to −20Nm by the PMSG control system and
-40
-30
-20
-10
0
10
0 1 2 3 4 5 6-30
-20
-10
-40
-30
-20
-10
0
10
0 1 2 3 4 5 6-30
-20
-10
Fig. 7: Experimental results for step changes in the stator resistance Rs of the PMSG: (a) Proposed PTC, and (b)traditional PTC.
the mechanical speed of the shaft is kept constantat 100 rad/s by the RSM control technique. It is clearfrom this figure that the weighting factor γ is playingan important role in the ripples that appeared in thecurrent waveform. Accordingly, the weighting factorγ = 0.8 is selected in the work.
The dynamic performance of the proposed PTCand traditional one is shown in Fig. 6. At the timeinstants t = 1.0 s and t = 3.0 s, step changes in thereference electro-magnetic torque T ⋆
e from 0Nm to−40Nm and then to −20Nm, respectively, havebeen applied to the PMSG control strategy. Themechanical speed of the shaft ωm is kept constantat 80 rad/s. It can been seen from Fig. 6 that thedynamic performance of the proposed PTC is sim-ilar to that of the traditional one. However, theproposed PTC requires approximately 15µs execu-tion time, while the traditional PTC requires approx-imately 35µs. Hence, the computational load is re-duced to 15
35×100% = 42% (i.e., a reduction by 58%).Furthermore, in the proposed PTC, no effort is re-quired for tuning of the weighting factor.
The robustness of the proposed PTC is also inves-tigated and compared with the traditional one. InFig. 7, the performance of the proposed PTC andtraditional one for ∓25% software step changes inthe stator resistance Rs of the PMSG is illustrated.The electro-magnetic torque Te is set to −30Nm
and the mechanical speed of the shaft ωm is keptconstant at 120 rad/s. According to that figure, both
control schemes (i.e. proposed and traditional PTC)show good robustness to variations of the stator re-sistance Rs of the PMSG.
Finally, the performance of the proposed PTC andtraditional one under variations of the stator induc-tance Ls of the PMSG is given in Fig. 8. Theelectro-magnetic torque Te is set to −25Nm andthe mechanical speed of the shaft ωm is kept con-stant at 90 rad/s. It can be observed that both con-trol techniques (i.e. proposed and traditional PTC)are sensitive to mismatches in the stator inductanceLs of the PMSG. This is because predicting thetorque/d-axis current and computing the referencevoltage vector are highly dependent on the parame-ters of the machine. However, both control systemsare still stable.
7. Conclusion
In this paper, a computationally efficient PTC tech-nique without weighting factors for PMSGs is pro-posed. The proposed PTC strategy is based on us-ing the d-axis current of the PMSG to be the secondcontrol variable beside the torque, which reduces(slightly) the calculation burden. Furthermore, in or-der to overcome the weighting factors tuning prob-lem in the cost function, the reference current ofthe q-axis is computed according to the referencetorque. Then, the reference VV is directly computedfrom the reference d- and q-axis currents using a
-40
-30
-20
-10
-10
0
10
0 1 2 3 4 5 6-25
-20
-15
-10
-5
-40
-30
-20
-10
-10
0
10
0 1 2 3 4 5 6-25
-20
-15
-10
-5
Fig. 8: Experimental results for step changes in the stator inductance Ls of the PMSG: (a) Proposed PTC, and (b)traditional PTC.
deadbeat function. Finally, in order to reduce thecomputational effort, the sector where the referenceVV is located is determined. Therefore, three eval-uations of the cost function are only required to findthe optimal VV. The performance of the proposedPTC technique is experimentally investigated andcompared with that of the conventional one. Theresults have shown that: 1) The calculation burdenof the proposed PTC strategy is significantly lowerthat of the traditional one, 2) the dynamic/steady-state performance of the proposed PTC techniqueis similar to that of the traditional PTC, and 3) boththe proposed and traditional PTC techniques aresensitive to variations of the machine parameters,in particular, the inductance of the stator.
8. Acknowledgment
This work is supported by the project AWESCO (H2020-ITN-642682) funded by the European Union’s Horizon2020 research and innovation programme under theMarie Sklodowska-Curie grant agreement No. 642682.
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