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Model predictive direct current control for multi-level converters

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Model predictive direct current control for multi-level converters
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  1 Model Predictive Direct Current Controlfor Multi-Level Converters Tobias Geyer,  Member, IEEE   Abstract —A model predictive current controller for multi-level inverter driving electrical machines is proposed that keepsthe stator currents within given bounds around their respectivereferences and balances the inverter’s neutral point potentialaround zero. The inverter switch positions are directly set bythe controller thus avoiding the use of a modulator. Admissibleswitching sequences are enumerated and a state-space model of the drive is used to predict the drive’s response to each sequence.The predicted short-term switching losses are evaluated andminimized. The concept of extrapolation and the use of boundsachieve an effective prediction horizon of up to 100 time-stepsdespite the short switching horizon. When compared to classicmodulation schemes such as pulse width modulation, for longprediction horizons, the switching losses and/or the harmonicdistortion of the current are almost halved when operating atlow pulse numbers, thus effectively resembling the steady-stateperformance of optimized pulse patterns. During transients thedynamic response time of the proposed controller is in the rangeof a few ms and thus very fast.  Index Terms —Model predictive control, current control,medium-voltage drive I. I NTRODUCTION In high power applications exceeding one megawatt multi-level inverters are typically used – rather than two-levelinverters – in order to reduce the rating of the semiconductorswitching devices, to minimize the harmonic distortions and toincrease the modulated voltage. The inverter must be operatedsuch that the desired three-phase load currents are produced.Several control methodologies are available to address thiscurrent control problem in three-phase voltage source invert-ers. As shown in the survey paper [1], the controllers can begrouped into linear and nonlinear control schemes.The most prominent representative of a linear controlmethodology is Field Oriented Control (FOC), which is for-mulated in a rotating orthogonal reference frame [2]. Two(orthogonal) control loops are used, typically with ProportionalIntegral (PI) controllers augmented with feedforward terms –one for the torque producing and one for the flux producingcurrent. A subsequent Pulse Width or Space Vector Modulator(PWM or SVM) translates the stator voltage reference signalsinto gating commands for the inverter [3]. Examples for non-linear current control schemes include hysteresis controllers,which typically directly set the inverter switch positions. In adrive setting the current control loop typically constitutes theinner loop within a cascaded control loop. On the machineside, the outer loop includes the torque and/or speed and theflux control loops, while on the grid side the active and reactivepower is controlled. T. Geyer is currently with the Department of Electrical and Computer Engi-neering, The University of Auckland, New Zealand; e-mail: t.geyer@ieee.org Recently, the power electronics community has started toinvestigate the concept of Model Predictive Control (MPC) [4],[5]. The roots of MPC can be traced back to the processindustry, where the origins of MPC were developed in the1970s [6]. The emerging field of MPC for three-phase voltagesource inverters can be divided into two categories. The firstone builds on FOC and replaces the inner (current) controlloop by MPC and keeps the modulator in place. Examplesfor this approach include [7] and [8]. In the second variety,MPC directly manipulates the inverter switch positions thussuperseding a modulator. For Neutral Point Clamped (NPC)inverters the latter scheme is available with a predictionhorizon of one as introduced in [9].This paper proposes an MPC based model predictive currentcontroller with very long prediction horizons in the range of 100 time-steps. Specifically, a Model Predictive Direct CurrentController (MPDCC) for multi-level inverter is proposed thatkeeps the stator currents within specified bounds around theirreferences, balances the inverter’s neutral point potential(s)around zero and minimizes either the inverter switching lossesor its switching frequency. The control problem is formu-lated in an orthogonal reference frame that can be eitherstationary or synchronously rotating. The formulation of thecurrent bounds in different reference frames is compared witheach other and with the bounds resulting from MPDTC. Amodulator is not required, since the gating signals are directlysynthesized by the controller.The key benefit of this approach is that the current controland the modulation problems are addressed in one computa-tional stage. As a result the current harmonic distortion andthe switching losses can be reduced at the same time whencompared to PWM. Indeed, at low switching frequencies, theresulting steady-state behavior is similar to the one obtainedby Optimized Pulse Patterns (OPP). Yet, during transients, avery fast current response time is achieved in contrast to OPPs,which tend to be applicable only in very slow control loops.This MPDCC scheme can be considered as an adaptationof Model Predictive Direct Torque Control (MPDTC) to thecurrent control problem. This is achieved by changing thecontrol objectives – namely, instead of controlling the torqueand flux magnitude the stator currents are controlled. MPDTCwas developed in early 2004, see [5] and [10], with predictionhorizons in the range of a few dozen, experimentally verifiedon a 2.5MVA drive in 2007 [11] and later generalized to en-able even longer prediction horizons [12]. Preliminary resultsof a MPDCC scheme for a two-level inverter based on theinitial MPDTC algorithm minimizing the inverter switching July 1, 2010 ECCE 2010  2 frequency and using relatively short prediction horizons werepresented in [13].II. P HYSICAL  M ODEL OF THE  D RIVE  S YSTEM Throughout this paper, we will use normalized quantities.Extending this to the time scale  t , one time unit correspondsto  1 /ω b  seconds, where  ω b  is the base angular velocity.  A. The  αβ  0  Reference Frame All variables  ξ  abc  = [ ξ  a  ξ  b  ξ  c ] T  in the three-phase system(abc) are transformed to  ξ  αβ 0  = [ ξ  α  ξ  β  ξ  0 ] T  in the orthogonal αβ  0  stationary reference frame through  ξ  αβ 0  =  P ξ  abc . Usingthe  αβ  0  reference frame and aligning the  α -axis with the a-axis, the following transformation matrix is obtained P   = 23  1  − 12  − 12 0 √  32  − √  32121212  .  (1)  B. Physical Model of the Inverter  As an illustrative example for a variable speed drive systemwith a multi-level inverter consider a three-level Neutral PointClamped (NPC) voltage source inverter driving an inductionmachine, as depicted in Fig. 1. The total dc-link voltage  V  dc over the two dc-link capacitors  x c  is assumed to be constant.Let the integer variables  u a ,  u b ,  u c  ∈ {− 1 , 0 , 1 }  denote theswitch positions in each phase leg – the so called phase states,where the values  − 1 , 0 , 1  correspond to the phase voltages − V   dc 2  , 0 ,  V   dc 2  , respectively. Note that in a three-level inverter 27different switch combinations exist. The actual voltage appliedto the machine terminals is given by  v αβ 0  = 0 . 5 V  dc  P u abc with  u abc  = [ u a  u b  u c ] T  .The neutral point potential  υ n  = 0 . 5( V  dc,lo − V  dc,up )  betweenthe two capacitors floats. In here,  V  dc,lo  and  V  dc,up  denote thevoltage over the lower and upper dc-link half, respectively.The neutral point potential changes when current is drawndirectly from it, i.e. when one of the switch positions is zero.Taking into account that the phase currents sum up to zero,i.e.  i sa  +  i sb  +  i sc  = 0 , it is straightforward to derive dυ n dt  = 12 x c | u abc | T  P  − 1 i s,αβ 0  ,  (2)where  i s,αβ 0  is the stator current expressed in the statorreference frame, and  | u abc |  = [ | u a | | u b | | u c | ] T  is the com-ponentwise absolute value of the inverter switch positions [5].To avoid a shoot-through direct switching between the upperand lower rails is prohibited.Switching losses arise in the inverter when turning thesemiconductors on or off and commutating the phase current.These losses depend on the applied voltage, the commutatedcurrent and the semiconductor characteristics. ConsideringIntegrated Gate Commutated Thyristors (IGCT), with the GCTbeing the semiconductor switch, the switch-on and switch-off losses can be well approximated to be linear in the dc-link voltage and the phase current. Yet for diodes, the reverserecovery losses are linear in the voltage, but nonlinear in thecommutated current. As shown in [12], [14], the switching V  dc x c x c NNNABC i s,abc IMFig. 1: Three-level neutral point clamped VSI driving an induction motor losses can be derived as a function of the switching transition,the commutated phase current and its polarity. C. Physical Model of the Machine The state-space model of a squirrel-cage induction machinein the stationary  αβ   reference frame is summarized hereafter.For the current control problem at hand it is convenient tochoose the stator currents  i sα  and  i sβ  as state variables. Thestate vector is complemented by the rotor flux linkages  ψ rα and  ψ rβ , and the rotor’s angular velocity  ω r . The model inputare the stator voltages  v α  and  v β . The model parameters arethe stator and rotor resistances  r s  and  r r , the stator, rotor andmutual reactances  x ls ,  x lr  and  x m , respectively, the inertia  J  ,and the mechanical load torque  T  ℓ , where the rotor quantitiesare referred to the stator circuit.The continuous-time state equations are [15], [16] i sα  +  τ  σ ′ di sα dτ   =  k r r σ τ  r ψ rα  +  k r r σ ω r ψ rβ  + 1 r σ v α  (3a) i sβ  +  τ  σ ′ di sβ dτ   =  k r r σ τ  r ψ r β  −  k r r σ ω r ψ rα  + 1 r σ v β  (3b) ψ rα  +  τ  r dψ rα dτ   =  − ω r τ  r ψ rβ  +  x m i sα  (3c) ψ rβ  +  τ  r dψ rβ dτ   =  ω r τ  r ψ rα  +  x m i sβ  (3d) τ  m  ·  dω r dτ   =  T  e  −  T  ℓ  ,  (3e)with the electromagnetic torque T  e  =  k r ( i sβ ψ rα  −  i sα ψ rβ ) .  (4)The deduced parameters used in here are the coupling factorof the rotor  k r  =  x m x r , the total leakage factor  σ  = 1  −  x m 2 x s x r ,the equivalent resistance  r σ  =  r s  +  k r 2 r r  and the leakagereactance  x σ  =  σx s , where  x s  =  x ls + x m  and  x r  =  x lr + x m .The deduced time constants include the transient stator timeconstant  τ  σ ′  =  σx s r σ , the rotor time constant  τ  r  =  x r r r and themechanical time constant  τ  m  = 1 /J  .III. C URRENT  C ONTROL  P ROBLEM The control problem is to regulate the stator currents aroundtheir references. During transients a high dynamic performanceis to be ensured, i.e. a short settling time in the range of afew ms. At steady state operating conditions the harmonicdistortion of the current is to be minimized so as to reducethe copper losses and thus the thermal losses in the stator July 1, 2010 ECCE 2010  3 i rip ,α              i     r     i    p  ,      β − δ  i − δ  i δ  i δ  i 00 (a) Current ripple bounds in  αβ   re-sulting from (6) − δ  i − δ  i δ  i δ  i 00 i rip ,a              i     r     i    p  ,      b (b) Current ripple bounds in  ab  and ac  resulting from (8) − δ  i − δ  i δ  i δ  i 00 i rip ,b              i     r     i    p  ,    c (c) Current ripple bounds in  bc  result-ing from (8) i rip ,α              i     r     i    p  ,      β -0.6-0.6-0.3-0.3000.30.30.60.6 (d) Torque and flux bounds translatedinto current ripple bounds in  αβ  Fig. 2: Bounds on the current ripple in  αβ  ,  ab ,  ac  and  bc , when imposing current bounds in  abc  or in  αβ  , respectively. The right most figure shows thecurrent ripple bounds in  αβ   resulting from the torque and flux bounds imposed in model predictive direct torque control winding of the machine. The current’s harmonic distortiondirectly relates to the current ripple, which is defined as thedeviation of the instantaneous current from its reference. Thusinstead of reducing the current harmonic distortion we can alsominimize the ripple current. The proportionality between theripple and the harmonic distortion will be shown in Sect. VI-C.With regards to the inverter the switching losses in thesemiconductors are to be minimized. An indirect way of achieving this is to reduce the device switching frequency.The inverter’s state(s) such as the neutral point potential hasto be balanced around zero.A suitable measure for the harmonic distortion of the currentis the Total Demand Distortion (TDD) I  TDD  =   0 . 5  h  =0  I  2 h I  nom ,  (5)in which the nominal current  I  nom  refers to the operatingcondition at nominal speed and load of the drive. The (har-monic) Fourier components  I  h ,  h  ≥  0 , can be differentiatedinto the fundamental current component  I  0  and the  h -thharmonic amplitude component  I  h . The harmonic distortionof the electromagnetic torque is defined accordingly.IV. F ORMULATION OF THE  S TATOR  C URRENT  B OUNDS The bounds on the stator currents can be imposed invaries manners. Assume symmetric bounds around the currentreference. Let  δ  i  denote the difference between the upper(lower) bound and the reference.The natural choice [1] is to impose upper and lower boundson the  abc  current of the form | i rip ,a | ≤  δ  i  ,  | i rip ,b | ≤  δ  i  ,  | i rip ,c | ≤  δ  i  ,  (6)where the ripple current in phase  a  is defined as  i rip ,a  = i s,a  −  i ref  ,a . The ripple currents in phase  b  and  c  are definedaccordingly. Using (1) and taking into account that the ripplecurrents are common mode free (the machine’s star point isnot connected), the constraints (6) can be translated from the abc  into the  αβ   frame. | i rip ,α | ≤  δ  i  ,  | i rip ,α |  + √  3 | i rip ,β | ≤  2 δ  i  (7)The set of ripple currents in  αβ   that meet (6) is depicted inFig. 2(a) as a gray polygon. The edges of the polygon arecalled facets. The facets are perpendicular to the  a ,  b  and  c -axes, respectively. The distance of the facets to the srcin isgiven by  δ  i . The  0 -component of the current ripple is alwayszero.Conversely, one might impose upper and lower bounds onthe currents in the  αβ   frame as proposed e.g. in [13]. | i rip ,α | ≤  δ  i  ,  | i rip ,β | ≤  δ  i  (8)This constraint is visualized in Fig. 2(a) as a red square.Translating the set imposed by (8) from  αβ   to  abc  yields anon-trivial shape. Fig. 2(b) shows the set in an orthogonalplane spanned by the  a  and  b -axis, which is the same as for ac , while Fig. 2(c) shows the set in the  bc  plane. The redpolygons in Figs. 2(b) and 2(c) refer to the constraint (6).It is obvious that the two constraint formulations (6) and(8) lead to different sets in  αβ   and  abc . The current harmonicdistortion relates to the ripple in  abc  rather than in  αβ  . Thus,from a TDD perspective, it is advantageous to impose theconstraint (6) rather than (8). This is confirmed by simulationresults, even though the difference amounts only to severalpercent and is thus fairly small. Since the machine model isformulated in  αβ   it is convenient to formulate the currentconstraints also in this reference frame. Therefore, the con-straint formulation (7), which is equivalent to (6), is adoptedfor MPDCC.On the other hand, in a model predictive direct torque andflux control setting, i.e. MPDTC, the stator flux vector is thekey figure to be controlled. Specifically, the angle betweenthe stator and rotor flux vectors determines the electromagnetictorque, while the stator flux’s magnitude is usually kept aroundits nominal value to keep the machine fully magnetized. Byimposing upper and lower bounds on the torque and the statorflux magnitude a target window results that defines the rippleof the stator flux vector. Due to the direct correspondencebetween the stator flux and the stator current, the stator flux’starget window can be translated into an equivalent window forthe stator current ripple in  αβ  . The latter is shown in Fig. 2(d).Since the bounds on the stator flux magnitude are typicallyasymmetric, the set of ripple currents is also asymmetric with July 1, 2010 ECCE 2010  4 Speedcontroller MPDCCMEncoder (optional) Minimization of cost functionPrediction of trajectories Dc-linkObserver  =~~ Fluxcontroller  Ψ r, ref  ω ref  i s, ref  i s u Ψ r ω r e - jδ δ  Fig. 3: Model predictive direct current control (MPDCC) for a multi-levelvoltage source inverter driving an electrical machine respect to the srcin. The curvature results from the bounds onthe stator flux magnitude. Note that in  αβ   this window rotatesaround the srcin.V. M ODEL  P REDICTIVE  D IRECT  C URRENT  C ONTROL As shown in Fig. 3, MPDCC constitutes the inner currentcontrol loop formulated in the stationary  αβ   reference frame.The current loop is augmented in a cascaded controller fashingby an outer loop that operates in the rotating  dq   frame andcomprises a flux and a speed PI controller with feedforwardterms.  A. Internal Controller Model MPC relies on an internal model of the physical drivesystem to predict the future drive trajectories, specifically thecurrent and neutral point trajectories.The overall state vector of the drive is chosen to be x  = [ i sα  i sβ  ψ rα  ψ rβ  υ n ] T  , the switch positions constitutethe input vector  u  =  u abc  = [ u a  u b  u c ] T  ∈ {− 1 , 0 , 1 } 3 , andthe stator current along with the neutral point potential is theoutput vector  y  = [ i sα  i sβ  υ n ] T  . The rotor speed is assumedto be effectively constant within the prediction horizon, whichturns the speed into a time-varying parameter. The predictionhorizon being in the range of a few ms, this appears tobe a mild assumption for medium-voltage drive applications.Nevertheless, including the speed as an additional state in themodel might be necessary for highly dynamic drives and/ordrives with a small inertia.Combining the motor model (3)–(4) with the invertermodel (2) and using the Euler formula, a discrete-time state-space model of the drive can be derived with the samplinginterval  T  s  = 25 µ s. The resulting state equation is bilinearin the input variable due to (2). The discrete-time model isomitted here due to space limitations, but it is conceptuallysimilar to the one in [10].  B. Generalized MPDCC Algorithm In MPDCC, the two stator current components are to be keptwithin given bounds around their respective references, whilethe neutral point potential is to be balanced around zero, seeFig. 3. For this, the inverter switch positions are directly set byMPDCC thus making a modulator obsolete. A machine and aninverter model is used to assess possible switching sequencesover a long prediction horizon. The switching sequence ischosen that minimizes the predicted inverter switching losses.Out of this sequence only the first gating signal (at the currenttime-instant) is applied.Starting at the current time-step  k , the MPDCC algorithmiteratively explores the tree of feasible switching sequencesforward in time. At each intermediate step, all switchingsequences must yield output trajectories that are either  fea-sible , or  pointing in the proper direction . We refer to theseswitching sequences as  candidate  sequences. Feasibility meansthat the output variable lies within its corresponding bounds;pointing in the proper direction refers to the case in whichan output variable is not necessarily feasible, but the degreeof the bound’s violation decreases at every time-step withinthe switching horizon. The above conditions need to hold componentwise , i.e. for all three output variables 1 .It is important to distinguish between the switching horizon(number of switching instants within the horizon, i.e. thedegrees of freedom) and the prediction horizon (number of time-steps MPC looks into the future). Between the switchinginstants the switch positions are frozen and the drive behavioris extrapolated until a hysteresis bound is hit. The concept of extrapolation gives rise to long prediction horizons (typically30 to 100 time-steps), while the switching horizon is very short(usually one to three). The switching horizon is composedof the elements ’S’ and ’E’, which stand for ’switch’ and’extrapolate’ (or more generally ’extend’), respectively. We usethe task ’e’ to add an optional extension leg to the switchinghorizon. For more details and visualizations about the conceptof the switching horizon and its elements ’S’, ’E’ and ’e’, thereader is referred to [12].At time-step  k , the generalized MPDCC algorithm com-putes the three-phase switch position  u ( k )  according to thefollowing procedure.1) Initialize the root node with the current state vector x ( k ) , the last switch position  u ( k − 1)  and the switchinghorizon. Push the root node onto the stack.2a) Take the top node with a non-empty switching horizonfrom the stack.2b) Read out the first element. For ’S’, branch on all feasibleswitch transitions. For ’E’, extend the trajectories eitherby extrapolation as detailed in [5] or by using theinternal controller model of Sect. V-A.2c) Keep only the switching sequences that are candidates.2d) Push these sequences onto the stack.2e) Stop if there are no more nodes with non-empty switch-ing horizons. The result of this are the predicted (candi-date) switching sequences  U  i ( k ) = [ u i ( k ) ,...,u i ( k  + n i − 1)]  over the variable-length prediction horizons  n i ,where  i  ∈ I   and  I   is an index set. 1 As an example, consider the case where the  α -current component isfeasible, the  β  -current component points in the proper direction and the neutralpoint potential is feasible. July 1, 2010 ECCE 2010  5 0 5 10 15 20 25 3000.20.40.60.81 Time (ms) (a) Electromagnetic torque 0 5 10 15 20 25 30−1.5−1−0.500.511.5 Time (ms) (b) Stator currents in  abc 0 5 10 15 20 25 30−101−101−101 Time (ms) (c) Switch positions in  abc Fig. 4: Dynamic response of model predictive direct current control during torque steps of magnitude 1pu. The torque reference with the torque response,the three-phase stator currents and the switch positions are shown versus the time-axis in ms. The rotor’s angular velocity is  ω r  = 0 . 6 pu, the current boundwidth is  δ i  = 0 . 12  and the switching horizon is ’eSESE’ 3) Compute for each (candidate) sequence  i  ∈ I   the associ-ated cost. If the switching frequency is to be minimized,consider  c i  =  s i /n i , which approximates the averageswitching frequency, where  s i  =   k + n i − 1 ℓ = k  || u i ( ℓ )  − u i ( ℓ  −  1) || 1  is the total number of switch transitionsin the switching sequence  U  i ( k ) , and  n i  is the corre-sponding sequence length. Conversely, if the losses aretargeted, the cost function  c i  =  E  i /n i  is used, where  E  i denotes the switching losses.4) Choose the switching sequence  U  ∗  =  U  i ( k )  with theminimal cost, where  i  = argmin i ∈I   c i .5) Apply (only) the first switch position  u ( k ) =  u ∗  of thissequence and execute the above procedure at the nexttime-step  k  + 1 .Alternatively, by adapting the drive model, MPDCC can alsobe formulated in a  dq   reference frame rotating synchronouslywith the rotor. In  dq   the current references are constant andso are the upper and lower bounds. Yet, the hexagon-shapedbounds, see Fig. 2(a), would rotate in the  dq   frame. A possiblesimplification would be to approximate the hexagon by acircle. Moreover, in  dq  , the voltage vectors depend on theangular position of the frame complicating the computationof the drive response in the MPDCC Step 2b).The controller’s computation time of one sampling intervalhas been neglected above. Using the internal controller modelof the drive and the previously chosen switch position, thisdelay can be easily compensated by translating the measure-ments one time-step forward. For more details, see [11].VI. P ERFORMANCE  E VALUATION As a case study, consider a three-level NPC voltage sourceinverter driving an induction machine as shown in Fig. 1.A  3 . 3 kV and  50 Hz squirrel-cage induction machine rated at 2 MVA is used as an example for a commonly used medium-voltage induction machine. The machine and inverter parame-ters are summarized in Table I. The semiconductors used areABB’s 35L4510 4.5kV 4kA IGCT and ABB’s 10H4520 fastrecovery diode. The pu system is established using the basequantities  V  B  =   2 / 3 V  rat  = 2694 V,  I  B  = √  2 I  rat  = 503 . 5 Aand  f  B  =  f  rat  = 50 Hz. As previously,  δ  i  denotes the widthof the bounds on the  abc  current components, which aresymmetric around the reference, where  δ  i  is equal to the upperbound minus the reference.  A. Transients At 60% speed steps of magnitude 1pu in the torquereference are applied to MPDCC. As shown in Fig. 4 a veryfast current and thus torque response is achieved limiting thelength of the transients to about 1.5ms. It is apparent fromthe control algorithm described in Sect. V that MPDCC issimilarly fast as deadbeat and hysteresis control schemes. Notethat excessive switching during the transients is avoided as canbe seen from Fig. 4(c).  B. Steady-State Operation At 60% speed and full torque closed-loop simulationswere run to evaluate MPDCC’s performance at steady-stateconditions. The key performance criteria here are the harmonicdistortions of the current and the torque, and the switchinglosses in the inverter. This performance evaluation is donefor switching horizons of varying length and various bounds.MPDCC is compared with two well-established modulationmethods: PWM/SVM and optimized pulse patterns (OPP).Specifically, a three-level regular sampled PWM is usedwith two triangular carriers, which are in phase (phase dis-position). It is generally accepted that for multi-level inverterscarrier-based PWM with phase disposition (PD) results in thelowest harmonic distortion. As shown in [17] – by adding a  Induction Motor  Voltage 3300V  r s  0.0108puCurrent 356A  r r  0.0091puReal power 1.587MW  x ls  0.1493puApparent power 2.035MVA  x lr  0.1104puFrequency 50Hz  x m  2.3489puRotational speed 596rpm  Inverter  Dc-link voltage 5200V  V   dc  1.930pu x c  11.769puTABLE I: Rated values (left) and parameters (right) of the drive July 1, 2010 ECCE 2010
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