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A Cyclodissipativity Condition for Power Factor Improvement under Nonsinusoidal Source with Significant Impedance

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A Cyclodissipativity Condition for Power Factor Improvement under Nonsinusoidal Source with Significant Impedance
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  A Cyclodissipativity Condition for Power Factor Improvementunder Nonsinusoidal Source with Significant Impedance Dunstano del Puerto-Flores, Romeo Ortega, and Jacquelien M. A. Scherpen  Abstract —The main contribution of this paper is an exten-sion of a recent result that reformulates and solves the powerfactor compensation for nonlinear loads under nonsinusoidalregime in terms of cyclodissipativity. In the aforementionedresult the generator was assumed to be ideal, that is, withnegligible impedance. In this work, we formulate the powerfactor compensation problem in a way that explicitly accountsfor the effects of a significant source impedance. I. I NTRODUCTION Recently, in [1] it has been established that the classicalproblem in electrical engineering of optimizing energy trans-fer from an alternating current (ac) source to a load withnon-sinusoidal (but periodic) source voltage is equivalentto imposing the property of cyclodissipativity to the sourceterminals. Since this framework is based on the cyclodissi-pativity property, see [2], the improvement of power factor(PF) is done independent of the reactive power definition,which is a matter of discussions in the power community,see [3].Using this framework the classical capacitor or inductorcompensators were interpreted in terms of energy equal-ization, see [1] for more details. We have presented anextension of the results in [4] where we considered arbitrarylossless linear time-invariant (LTI) filters, and proved that forgeneral lossless LTI filters the PF is reduced if and only if acertain equalization condition between the weighted powersof inductors and capacitors of the load is ensured. Althoughthe aforementioned results were obtained by consideringnonlinear loads, the generator was assumed to be ideal,that is, with negligible impedance. However, in practice thesource impedance is often significant.We consider the energy transfer from a known voltagesource  v s  to a given fixed load  i ℓ , see Figure 1. Thestandard approach to improve the PF is to place a losslesscompensator,  Y  c , between the source and the load. ThePF compensation configuration considered in the paper isdepicted in Figure 2.In this work, our task is to formulate the power factor com-pensation problem in a way that explicitly accounts for theeffects of a non-negligible source impedance,  Z  s , on the loadvoltage and current. We prove that cyclodissipativity providesa rigorous mathematical framework useful to analyze and J.M.A. Scherpen and D. del Puerto-Flores are with the Deparmentof Discrete Technology and Production Automation, RijksuniversiteitGroningen, Nijenborgh 4, 9747 AG, Groningen, The Netherlands. (e-mail: { j.m.a.scherpen,d.del.puerto.flores } @rug.nl )R. Ortega is with the Laboratoire des Signaux et Syst`emes,Plateau de Moulon, 91192, Gif-sur-Yvette, cedex, France. (e-mail: ortega@lss.supelec.fr ) design power factor compensators for general nonlinear loadsoperating in nonsinusoidal regimes with significant sourceimpedance.First, we briefly review our result from [4]. Then, inSection IV we give a cyclodissipativity characterization of the lossless LTI PF Compensation problem with sourceimpedance. Next we provide a geometric characterization inSection V and we end with an example in Section VI.II. P OWER  F ACTOR  I MPROVEMENT  A. Framework  i s Z  s v us  i ℓ v s Fig. 1. Power delivery system with significant source impedance. i s i c Z  s Y  c v cs  i ℓ v s Fig. 2. Parallel load compensation in a power delivery system withsignificant source impedance. We consider the energy transfer from an  n -phase acgenerator to a load where the source is not assumed to beideal, but has a significant impedance, see Figure 1. Allsignals are assumed to be periodic and have finite power,that is, they belong to the space L n 2  =  x  : [0 ,T  )  → R n :   x  2 := 1 T     T  0 | x ( τ  ) | 2 dτ <  ∞  where  ·  is the rms value and  |·|  is the Euclidean norm.We also define the inner product in  L n 2  as  x,y   := 1 T     T  0 x ⊤ ( t ) y ( t ) dt. 49th IEEE Conference on Decision and ControlDecember 15-17, 2010Hilton Atlanta Hotel, Atlanta, GA, USA 978-1-4244-7746-3/10/$26.00 ©2010 IEEE3978  The process of power factor correction is an attempt toreduce the apparent voltamperes of a load to the value of theaverage power consumed. The universally accepted definitionof PF is given as [5]:  Definition 1 (Power factor):  Consider the power deliverysystems of the Figure 1. The PF of an AC electric powersystem is defined by PF   :=  P S  ,  (1)where P   :=   v us ,i s  , S   :=   v us  i s   (2)are the active (real) power, 1 and the apparent power, respec-tively.From (2), it follows that  P   ≤  S  . Hence  PF   ∈  [ − 1 , 1]  is adimensionless measure of the energy-transmission efficiency.Cauchy–Schwartz also tells us that a necessary and sufficientcondition for the apparent power to equal the active poweris that  v s  and  i s  are collinear. If this is not the case,  P < S  and compensation schemes are introduced to maximize thePF.The condition for unity power factor is that the inputcurrent to a systems is proportional at all times to theinstantaneous supplied voltage.  B. The Power Factor compensation problem The PF compensation configuration considered in thepaper is depicted in Figure 2, where  Y  c  :  L n 2  → L n 2  is theadmittance operator of the compensator. That is, i c  =  Y  c v cs where  i c  ∈ L n 2 , is the compensator current. In the simplestLTI case the operator  Y  c  can be described by its admittancetransfer matrix, which we denote by  ˆ Y  c ( s )  ∈  R n × n ( s ) . Wemake the following fundamental assumption.Following standard practice, we consider only losslesscompensators, that is,  i c ,v cs   = 0 ,  ∀ v cs  ∈ L n 2 .  (3)We recall that, if  Y  c  is a passive LTI LC-network, losslessnessimplies Re { ˆ Y  c (  jω ) }  = 0 .  (4)for all ω  ∈ R for which  jω  is not a pole of   ˆ Y  c  and all alternatezeros and poles are simple and lie on the imaginary axis andwhere  Re { ˆ Y  c (  jω ) }  is the real part of the admittance transfer ˆ Y  c (  jω ) , see [6].  Definition 2 (Power factor improvement):  Given a  n -phase source voltage  v s ( t ) , a linear time-invariant ( n -phase)source impedance  Z  s  :  L n 2  → L n 2 , and a fixed load current i ℓ , as in Figure 2, power-factor improvement is achievedwith the lossless compensator  Y  c  if and only if  PF > PF  u  :=   v us ,i ℓ  v us  i ℓ   (5) 1 Also called average power [5]. v s  N i s Zs  v p i p Fig. 3. Circuit schematic of an  n -phase non-ideal generator connected toa  n -port (possible nonlinear and time varying) load where  PF  u  denotes the uncompensated power factor, that is,the value of   PF   with  Y  c  = 0  and  i s  =  i ℓ , and, by Kirchhoff’sVoltage Law (KVL), the uncompensated voltage is v us  =  v s  − Z  s i ℓ .  (6)III. PF  COMPENSATION AND CYCLODISSIPATIVITY :I DEAL  C ASE The framework to be discussed carries abstract powerconnotations (as does the term itself). This is derived fromthe interpretation of the supply rate as an input power. Toformulate our results we need the following.  Definition 3 (Cyclodissipative system, [2]):  Given a map-ping  w  :  L n 2  ×L n 2  →  R . The  n -port system of Figure 3 iscyclodissipative with respect to the supply rate  w ( v  p ,i  p )  if and only if     T  0 w ( v  p ( t ) ,i  p ( t )) dt >  0 .  (7)for all  ( v  p ,i  p )  ∈ L n 2  ×L n 2 .  A. Cyclodissipative Framework for Power Factor Compen-sation To place or results in context, and make the paper self–contained, we recall the following results from [1]. Assume v s  is ideal, i.e.,  Z  s  = 0 . v s  i s  i c   i  l   Y  c   Y   l   Fig. 4. Parallel load compensation in a power delivery system withnegligible source impedance. Proposition 4 ([1]):  Consider the system of Figure 4 andfixed  Y  ℓ . The compensator  Y  c  improves the PF if and onlyif the system is cyclodissipative with respect to the supplyrate w ( v s ,i s ) := ( Y  ℓ v s  + i s ) ⊤ ( Y  ℓ v s  − i s ) .  (8)The next result follows from Proposition 4 and it character-izes the set of all compensators  Y  c  that improve the power-factor for a given  Y  ℓ . 3979  Corollary 5 ([1]):  Consider the system of Figure 4 Then Y  c  improves the PF for a given  Y  ℓ  if and only if   Y  c  satisfies 2  Y  ℓ v s ,Y  c v s  +  Y  c v s  2 <  0 ,  ∀ v s  ∈ L n 2 .  (9)Dually, given  Y  c , the PF is improved for all  Y  ℓ  that satisfy(9).  B. Compensation with Lossless LTI Compensator equatesWeighted Power Equalization. In this section we review the results presented in [4], whichextended the results in [1]. Similarly to [1], the class of RLC circuits that we consider as load models consists of possibly nonlinear lumped dynamic elements ( n L  inductors, n C   capacitors) and nonlinear static elements ( n R  resistors).Capacitors and inductors are defined by the physical lawsand constitutive relations [7]: i C   = ˙ q  C  , v C   =  ∇ H  C  ( q  C  ) ,  (10) v L  = ˙ φ L , i L  =  ∇ H  L ( φ L ) ,  (11)respectively, where  i C  ,  v C  ,  q  C   ∈  R n C are the capacitorscurrents, voltages and charges, and  i L ,  v L ,  φ L  ∈ R n L are theinductors currents, voltages and flux–linkages,  H  L  : R n L → R  is the magnetic energy stored in the inductors,  H  C   : R n C →  R  is the electric energy stored in the capacitors,and  ∇  is the gradient operator. We assume that the energyfunctions are twice differentiable. For linear capacitors andinductors H  C  ( q  C  ) = 12 q  ⊤ C  C  − 1 q  C  , H  L ( φ L ) = 12 φ ⊤ L L − 1 φ L , respectively, with  L  ∈  R n L × n L ,  C   ∈  R n C × n C . To avoidcluttering the notation we assume  L,C   are diagonal matrices.Finally, we distinguish between two sets of nonlinear staticresistors:  n R i  current–controlled resistors and  n R v  voltage–controlled resistors, for which the characteristic are given by v R i  = ˆ v R i ( i R i ) ,  and  i R v  =ˆ i R v ( v R v ) , respectively, where  i R i ,  v R i  ∈  R n Ri  are the currents, volt-ages of the current-controlled resistors, and  i R v ,  v R v  ∈ R n Rv are the currents, voltages of the voltage-controlled resistors,with  n R  =  n R i  + n R v .Recalling the definition of real power (2) we introduce thefollowing.  Definition 6 (Weighted (real) power):  Given a compen-sator admittance  Y  c  the weighted (real) power of a single–phase circuit with port variables  ( v,i )  ∈ L 2  ×L 2  is givenby P  w :=   Y  c v,i  .  (12)For instance, if   Y  c  is LTI P  w = ∞  k = −∞ ˆ Y  c [ k ]ˆ V  [ k ]ˆ I  ∗ [ k ]  (13)where  ˆ V  [ k ] ,  ˆ I  [ k ]  are the  k -th spectral lines of   v  and  i ,respectively, and  ˆ Y  c [ k ] := ˆ Y  c ( kω 0 ) , with  ω 0  :=  2 πT   . Thatis,  P  w is the sum of the power components of the circuitmodulated by the frequency response of   Y  c —hence the useof the “weighted” qualifier. 2 The aforementioned definitionmotivates the next result. Proposition 7:  Consider the system of Figure 2 with  n  =1 , 3 a full nonlinear RLC load and a fixed LTI losslesscompensator  Y  c  with admittance transfer function  ˆ Y  c (  jω ) which has a zero at the srcin.i) PF is improved if and only if  12 V  ws  + n L  q =1 P  wL q  + n C  q =1 P  wC  q  <  0  (14)where  V  ws  is the rms value of the filtered voltagesource, that is, V  ws  :=  || Y  c v s || 2 = ∞  k =1 | ˆ Y  c ( k )ˆ V  s ( k ) | 2 and P  wC  q  := ∞  k = −∞ ˆ Y  c [ k ]ˆ V  C  q [ k ]ˆ I  ∗ C  q [ k ] P  wL q  := ∞  k = −∞ ˆ Y  c [ k ]ˆ V  L q [ k ]ˆ I  ∗ L q [ k ] , are the weighted powers of the  q  –th inductor andcapacitor, respectively.ii) Condition (14) may be equivalently expressed as  ( 1  p Y  c ) v L , ∇ 2 H  L v L  −  i C  , ( 1  p Y  c ) ∇ 2 H  C  i C   >  12 V  ws (15)where  p  :=  ddt .iii) If the inductors and capacitors are linear their weightedpowers become P  wC  q  := 2 ω 0 ∞  k =1  k  I  m { ˆ Y  c [ k ] } n C  q =1 C  q | ˆ V  C  q [ k ] | 2  P  wL q  :=  − 2 ω 0 ∞  k =1  k  I  m { ˆ Y  c [ k ] } n L  q =1 L q | ˆ I  L q [ k ] | 2  . iv) Furthermore, the results i-iii can be extended for ageneral LTI lossless compensator, if the resistors of the load are linear time-invariants.Condition 14 indicates that the power factor improvementif and only if a certain equalization condition between theweighted powers of compensator and load is ensured. Wenow continue with extending these result to the non-idealsource case.IV. A C YCLODISSIPATIVITY  C ONDITION FOR  P OWER F ACTOR  I MPROVEMENT : N ON - IDEAL  C ASE Although the problem at hand is posed as a problem innetworks, it can be equally well interpreted as a feedback problem; the circuit of Fig. 2 is represented by the systemof Fig. 5, which consists of two systems in a feedback loop. 2 Since the spectral lines of real signals satisfy  ˆ F  [ − k ] = ˆ F  ∗ [ k ] , theweighted power is a real number. 3 This condition is imposed, without loss of generality, to simplify thepresentation of the result. 3980  Specifically, the inputs are  v s  and  i ℓ  and the outputs are i s  and  v cs , and products are related with the instantaneousdelivered power by the source  i ⊤ s  v s  and the instantaneousinput power into the load,  i ⊤ ℓ  v cs . Z  s  Y  c  v  s c  v  s   i  c  i  s   i   l  v  zs  Fig. 5. Feedback configuration Proposition 8:  Consider the system of Figure 2. Givena  n -phase source voltage  v s ( t ) , a linear time-invariant ( n -phase) source impedance  Z  s  and a fixed load current  i ℓ . Thecompensator  Y  c  improves the PF if and only if the system hasfinite gain and is cyclodissipative with respect to the supplyrate w ( i s ,i ℓ ) :=  δ  2 i ⊤ ℓ  i ℓ  − i ⊤ s  i s .  (16)for all  ( i s ,i ℓ )  ∈ L n 2  ×L n 2 , where  δ   is the upper gain bound 4 and is given by δ   =   v cs ,i ℓ  v cs  ( I   + Z  s Y  c ) v cs  ( I   + Z  s Y  c ) v cs ,i ℓ  ,  (17)with  1  < δ <  ∞ . Proof:  From Kirchhoff’s Current Law (KCL), we have i s  =  i c  + i ℓ , and using KVL, v cs  =  v s  − Z  s ( i c  + i ℓ ) ,  (18)Substituting (19) into (18), we obtain v us  =  v cs  + Z  s i c  (19)From the definition of power factor and the lossless conditionof the compensator, we have PF   :=   v cs ,i ℓ  v cs  i s   (20)And, we define ˜ α  :=   v cs ,i ℓ  v cs   .  (21) PF  u  is given by PF  u  :=   v us ,i ℓ  v us  i ℓ  and by using (19) and  i c  =  Y  c v cs , v us  =  v cs  + Z  s Y  c v cs  = ( I   + Z  s Y  c ) v cs , then, PF  u  =   ( I   + Z  s Y  c ) v cs ,i ℓ  ( I   + Z  s Y  c ) v cs  i ℓ  ,  (22) 4 See Table 1 of [8], Definition 2.1 of [9], and Definition 2 of [10]. and we define: α  :=   ( I   + Z  s Y  c ) v cs ,i ℓ  ( I   + Z  s Y  c ) v cs   .  (23)From Definition (5), we conclude that  PF > PF  u  if andonly if   v cs ,i ℓ  v cs  i s   >   ( I   + Z  s Y  c ) v cs ,i ℓ  ( I   + Z  s Y  c ) v cs  i ℓ  , or, by (21) and (23), the inequality becomes  i s   <  ˜ αα  i ℓ  ,  (24)with  δ   :=  ˜ αα . If   Y  c  = 0 , i.e., the uncompensated case, from(17) we have that  δ >  1  and because the fact that  δ   dependsonly on bounded signals,  i ℓ  and  v cs , and the operators  Z  s  and Y  c  are LTI, we can conclude that  δ <  ∞ .  Remark 9:  The results of Proposition 5 in [1] are aparticular case of Proposition 8 assuming an ideal source,i.e., the source impedance  Z  s  = 0  and, from (17), we have δ   = 1 .The next corollary of this result is the characterization of allcompensators that improve the power factor. Corollary 10:  Consider the system of Figure 2. Then  Y  c improves the PF for a given  i ℓ  if and only if   Y  c  satisfies  i c  2 + 2  i c ,i ℓ   <  ˜ αα  2 − 1   i ℓ  2 ,  ∀ v s ,i ℓ  ∈ L n 2 . (25) Proof:  Using the fact  i s  =  i c  +  i ℓ , and the square of (24),  i s  2 <  ˜ αα  2  i ℓ  2  i c  2 + 2  i c ,i ℓ  +  i ℓ  2 <  ˜ αα  2  i ℓ  2  i c  2 + 2  i c ,i ℓ   <  ˜ αα  2 − 1   i ℓ  2 . and, from  i c  =  Y  c v cs , then we have  Y  c v cs  2 + 2  Y  c v cs ,i ℓ   <  ˜ αα  2 − 1   i ℓ  2 (26)  Remark 11:  From the feedback configuration under con-sideration, see Fig. 5. The interconnected system is cy-clodissipative with respect to (16) if the compensator  Y  c  iscyclodissipative with respect to the supply rate function w c ( i ℓ ,i c ) :=  − i ⊤ c  i c  − 2 i ⊤ c  i ℓ  − (1 − δ  )(1 + δ  ) i ⊤ ℓ  i ℓ . where the input is  i ℓ  and  i c  is the output.V. G EOMETRICAL INTERPRETATION Referring to Fig. 6 we have a geometric interpretation of power factor compensation. Fig. 6 depicts the vector  v cs ,  v us , i s ,  i c  and  i ℓ . The angles  β   and  β  u  are defined as β   := cos − 1 PF, β  u  := cos − 1 PF  u , 3981  or,  β   = ∠ ( v cs ,i s )  and  β  u  = ∠ ( v us ,i ℓ ) . Then, it is clear fromFig. 6 that  PF > PF  u  if only if   β < β  u .From (17), by assuming that   v cs ,i ℓ   >  0  and   ( I   + Z  s Y  c ) v cs ,i ℓ   >  0 , then we have that  1  < δ <  ∞ . v  s c  v  s u  Z  s   v  s u  Y  C  i  c  i  s  ~ u   l  i  Fig. 6. Geometric interpretation of power factor compensation.  Remark 12:  The projection of   i ℓ  onto  v cs  is the vectordenoted and defined by proj( i ℓ ,v cs ) :=   v cs ,i ℓ  v cs  2  v cs , with magnitude ˜ α  :=   v cs ,i ℓ  v cs   .  Remark 13:  Consider the projection of   i ℓ  onto  v us  is thevector denoted and defined by proj( i ℓ ,v us ) :=   v us ,i ℓ  v us  2  v us , with magnitude α  :=   v us ,i ℓ  v us   , and with  v us  = ( I   + Z  s Y  c ) v cs , we obtain α  :=   ( I   + Z  s Y  c ) v cs ,i ℓ  ( I   + Z  s Y  c ) v cs   . VI. E XAMPLE In this section we present an example that illustrate someof the points discussed in the paper.Consider the three-phase distribution system of Fig 7.A linear, R-L load per phase, which admittance  Y  ℓ i  with i  ∈ { 1 , 2 , 3 } , is star-connected with a set of star-connected,compensated capacitors  ˆ Y  c i ( s ) =  C  c i s , with  i  ∈ { 1 , 2 , 3 } ,per phase. Assuming balanced, three-phase operation, withno d.c. component of current, the problem can be representedby the single-phase equivalent circuit in the Fig. 8, see[11].  v s ( t ) ,  v o ( t )  are the supply and load instantaneousvoltages per phase. The RMS voltage  ˆ V  s  is maintainedconstant at  33  kV. The distribution line has the followingdata: 3-phase, 20 miles of 336.4 MCM 5 , 26/7 ACSR 6 with14 ft. conductor spacing,  R  = 0 . 278 Ω  /mile/conductor, X  L  = 0 . 516 Ω  /mile/conductor. The R-L load per phase isassumed to be lumped resistance in series with lumped, pureinductance with 10 MV-A, 0.65 PF lagging at  50  Hz. n r  s1 r  s2  r  s3  L s1 L s2  L s3  Y   l 1  Y   l 2 Y   l 3 n v  s1 v  s2  v  s3  C  1  C  2   C  3  n  Fig. 7. Three-phase distribution system in parallel with a linear, balancedR-L load. v  s  r  s  R  l  L s  L  l  C v  o i  s  i  c  i   l  Fig. 8. Single-phase equivalent circuit of the three-phase system of Figure7. Now, consider the per-phase equivalent circuit, Fig. 8. Theimpedance line is  Z  s  = 20(0 . 278 +  j 0 . 516) Ω , the linearload is  Y  ℓ  = 1 /Z  ℓ  with  Z  ℓ  = 20(0 . 278 +  j 0 . 516) Ω , andthe uncompensated total impedance is  Z  uT   =  Z  s  +  Z  ℓ . Theimpedance of the compensator is  Z  c ( C  ) =  1 jωC  , then thecompensated total impedance Z  cT  ( C  ) =  Z  s  +  Z  c Z  ℓ Z  c  + Z  ℓ . Condition (24) helps us to obtain the parameters for a givencompensator  Y  c , i.e., the capacitance for this example, suchthat the power factor is improved. Where, the bounded gainis δ  ( C  ) = ℜ{ Z  cℓ }   ˆ V   s √  31 √  | Z  cT  |  2 ℜ{ Z  ℓ }   ˆ V   s √  31 √  | Z  uT  |  2ˆ V   s √  3 | Z  ℓ || Z  uT  | ˆ V   s √  3 | Z  cℓ || Z  cT  | ,  (27)the rms value of the input current is  i s ( C  )   =ˆ V  s √  31 | Z  cT  | , 5 The equivalent cross sectional area is 336,400 circular mils. 6 An Aluminum Conductor Steel Reinforced (or ACSR) cable with 26aluminum conductors and a core of 7 steel conductors. 3982
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