Load Static Models for Conservation Voltage Reduction in the Presence of Harmonics

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1 Energy and Power Engneerng, 06, 8, 6-75 Publshed Onlne February 06 n ScRes. Load Statc Models for Conservaton Voltage Reducton n the Presence of Harmoncs Wllam Douglas Caetano, Patríca Romero da Slva Jota * Electrcal Engneer Department, Federal Centre of Technologcal Educaton (CEFET-MG), Belo Horzonte, Brazl Receved December 05; accepted 3 February 06; publshed 6 February 06 Copyrght 06 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract The Conservaton Voltage Reducton (CVR) s a technque that ams to acheve the decrease of power consumpton as a result of voltage reducton. The customer s suppled wth the lowest possble voltage level compatble wth the stpulated level by the regulatory agency. Internatonal Standards ANSI C and IEEE std specfy the range of supply voltage to electroncs equpment from 0.9 to.05 pu of nomnal voltage. To analyse the CVR effect n dstrbuton systems wth dfferent load characterstcs (resdental, commercal, ndustral or a combnaton of these), mathematcal load models are used. Typcally, these equpment/load models are used to analyse load aggregaton wthout any consderaton of ts nonlnearty characterstcs. Amng to analyse the nonlnear characterstcs and ts consequences, ths paper presents a dscusson of the neglected varables as well as the results of a set of urements of nonlnear loads. Dfferent mathematcal models are appled to obtan them for each load. Usng these models the load aggregaton s evaluated. It s presented that although the models show adequate results for ndvdual loads, the same does not occur for aggregated models f the harmonc contrbuton s not consdered. Consequently, to apply the load model n CVR t s necessary to consder the harmoncs presence and the model has to be done usng only the fundamental frequency data. The dscusson about the causes s done and the models are compared wth the urements. Keywords CVR Load Model n the Presence of Harmoncs, Conservaton Voltage Reducton, Load Model Aggregaton n the Presence of Harmoncs * Correspondng author. How to cte ths paper: Caetano, W.D. and da Slva Jota, P.R. (06) Load Statc Models for Conservaton Voltage Reducton n the Presence of Harmoncs. Energy and Power Engneerng, 8,

2 W. D. Caetano, P. R. da Slva Jota. Introducton Conservaton Voltage Reducton (CVR) s a method of energy reducton consumpton resultng from a reducton of feeder voltage []. Accordng to Schneder [] whle n North Amerca there are numerous CVR systems deployed n dstrbuton feeders, the majorty of the results publshed are based on emprcal feld urements. Consequently, t s not possble to extrapolate the results of ths technology on other dstrbuton feeders. To obtan a better evaluaton, t s necessary to use load models and analyse the combnaton of these models n smulaton processes. The load model for CVR analyss s a mathematcal representaton of the relatonshp between the actve and reactve power consumed and load feed voltage; n other words, t descrbes the load behavor when t s connected on power grd consderng the voltage varaton. Such models can be done wthout thermal cycles or statc [] [] where the power at each nstant, t, s expressed as functon of voltage and frequency at the same nstant = (, ), Q f ( V, f ) PL fp V f L = Q ; or dynamc, where the power s expressed as functon of voltage and frequency both at past and present tme []. Loads wthout thermal cycles consume energy n a tme-nvarant manner, wth the excepton of voltage varatons. Specfcally, there s no control feedback loop. As an example, a lght bulb wll consume energy when turned on, as a functon of voltage n a fxed manner. In contrast, a load wth a thermal cycle, such as a water heater, wll have a varyng duty cycle dependent on the supply voltage, Schneder []. The load model, accordng to Prce et al. [], may brng benefts n plannng studes, n electrcal power systems operaton, resultng n, for example, reducton of system modfcatons and equpment nvestments, and system preventng emergences due to better operatng lmts. Some researchers found n lterature demonstrate the mportance of load model applcaton on Conservaton Voltage Reducton (CVR) []-[]. In 99, a research developed by Electrc Power Research Insttute (EPRI) [3] presented tests n order to verfy the reducton voltage effects n the effcency of loads exstng n resdental, commercal and ndustral sectors. Such tests were dvded n two parts. In the frst part, data were collected to model energy consumpton as a functon of supply voltage. In the second part, the collected data were used to model the loads. In 989, a feld test was performed n Snohomsh [4] n four crcuts connected to three substatons whose loads were predomnantly resdental and commercal, where there were alternatng cycles wth nomnal voltage and low voltage every 4 hours. In each substaton, power meters were nstalled to collect actve power, reactve power and voltage n each phase of the test crcut, where the data were collected every 5 mnutes. The test resulted n a reducton of energy consumpton. However the result was not consdered optmal because each substaton at each nstant of tme had dfferent loads connected. It s evdent that the load model s needed for predctng energy saved as a result of applyng the CVR method. In 00, the study descrbed n Schneder [] presented the CVR applcaton n some feeders, where the statc loads were modeled by a polynomal model (ZIP), and the dynamc loads were modeled by an equvalent thermal parameter (ETP) model. The dynamc loads used a more complex model due to the control system n each load that determnes the moment when t s energzed and for how long. None of those publcatons consder the nonlnear characterstcs of the load. The contrbuton of the harmoncs n the load behavor has to be consdered n the analyss. Ths paper evaluates three model statc load technques as used n lterature, wthout any consderaton of nonlneartes and ncludng ther analyses. Ths paper s dvded as followng: Frstly three CVR model technques were presented. These technques can be appled consderng or not the harmoncs components presence. Then, n the second part the approach dfferences were shown. To obtan the model usng the ured data, t was presented the parameter estmaton technque. It s desred to mantan the physcs aspects of the model; then the ellpsod algorthm (usng restrctons) and ts applcaton were presented. Fnally the used methodology was shown as well as the results. The comparson has been done and the conclusons were shown.. End-User Load Modelng Statc Loads End-user load models are mathematcal functons used to descrbe the behavor of commercal or resdental loads consderng the actve and reactve power as functon of both varables: voltage and sometmes frequency. The statc models used to represent the load are performed n two man types: by an exponental model and by polynomal models, also known as ZIP models [] [] [5] [7]. 63

3 W. D. Caetano, P. R. da Slva Jota.. Exponental Model EXP In exponental models the relatonshp between the consumed power and the feed voltage s gven by an exponental functon, as n Equaton () for the actve power and n Equaton () for the reactve power [] [5] [0]- []: α V P = P0 V o β V Q = Q0 V o where α and β parameters descrbe the load behavor. When both coeffcents α and β are equal to zero, t means that the load behaves as Constant Power. In case of both α and β are equal to one, the load behaves as Constant Current; and fnally f both coeffcents are equal to two, the load behaves as Constant Impedance... ZIP Model (ZIP) In polynomal models, ZIP model, the load s represented by followng some physcal characterstcs. Some equpment, such as electronc ballasts, have constant power consumpton under voltage varaton. On the other hand, others behave as a combnaton of two or three of the followng characterstcs: Constant Impedance (Z), Constant Current (I) and Constant Power (P). The expressons for actve and reactve power (P, Q ) of the ZIP model are presented by Equatons (3) and (4) respectvely. For both Equatons, the coeffcents have to obey the constrants presented n Equaton (5). The relatonshp between power consumpton and voltage magntude s gven by a polynomal Equaton [] [5] [7] [0] []: P P P I Z V V = 0 p p p Vo Vo Q Q P I Z V V = 0 q q q Vo Vo Zp Ip Pp = Zq Iq Pq = where: P : Actve power consumpton of the th load. Q : Reactve power consumpton of the th load. P o : Actve power consumpton at nomnal voltage. Q o : Reactve power consumpton at nomnal voltage. V : Actual termnal voltage. V o : Nomnal termnal voltage. Z pq, : Fracton of load that s Constant Impedance. I pq, : Fracton of load that s Constant Current. P : Fracton of load that s Constant Power. pq,.3. Modfed ZIP Model (ZIP_mod) Schneder et al. [] proposed a new ZIP model where, besdes the nformaton about the Constant Impedance, Constant Current and Constant Power, there s also nformaton about the phase angle of each one of these components. In ths case, both actve and reactve Equatons are coupled. The actve and reactve power calculus s gven n Equatons (6) and (7) respectvely, obeyng the constrant n Equaton (8): () () (3) (4) (5) 64

4 W. D. Caetano, P. R. da Slva Jota V V P= SZ cos Z SI cos I SP cos P n % θ n % θ n % θ Vo Vo (6) V V Q= SZ sn Z SI sn I SP sn P n % θ n % θ n % θ Vo Vo Z% I% P% = (8) where: P : Real power consumpton of the th load. Q : Reactve power consumpton of the th load. V : Actual termnal voltage. V o : Nomnal termnal voltage. S n : Apparent power consumpton at nomnal voltage. Z % : Fracton of load that s Constant Impedance. I % : Fracton of load that s Constant Current. P % : Fracton of load that s Constant Power. Z θ : Phase angle of the Constant Impedance component. I θ : Phase angle of the Constant Current component. P θ : Phase angle of the Constant Power component. In any ZIP model, when the load s represented as Constant Impedance, t means that the power vares drectly wth the square of the voltage; n the case of the Constant Current, the power vares drectly wth the voltage; and, lastly, f the load s modeled as Constant Power, the power does not vary when the voltage vares []. 3. Analysng the Harmoncs Presence and Its Effect n the Model None n all presented models s mentoned about harmonc components n any reference although ts presence affects the data. Once urements are used to obtan the parameters for each model, and as the components suffer nfluences from harmoncs, t s necessary to dscuss how to consder ts nfluence before start usng the data. 3.. For Snusodal Voltage and Current The apparent power ured by the equpment gves us two parcels called P and Q. For snusodal source supplyng a lnear load, these parcels are [3]: where: V: RMS voltage. V : Fundamental voltage. V : Fundamental current. S: Apparent power. P : Fundamental actve power. Q : Fundamental reactve power. P M : Measured actve power. Q M : Measured reactve power. 3.. For Nonsnusodal Voltage and Current M M S = VI = V I = P Q = P Q (9) Consderng nonsnusodal source supplyng a nonlnear load, however, the apparent power assumes a dfferent value ncludng the harmoncs effect. The ured actve power P M now has also the contrbuton of the harmonc actve power P H. Smlarly the ured reactve power Q M s not only composed by Q but also by a sum of other parcels, namely D V, D I and D H. Ths ured power should no longer be called reactve power, but nonactve power, [3]. The demonstraton s presented below: (7) 65

5 W. D. Caetano, P. R. da Slva Jota consequently, = = H H = H H H H S VI V V I I VI VI V I V I = H H H H S P Q VI V I P D S = P Q D D P D V I H H S = P N where: V : Fundamental voltage. V H : Harmonc voltage. I : Fundamental current. I H : Harmonc current. S: Apparent power. P : Fundamental actve power. Q : Fundamental reactve power. P M : Measured actve power. Q M : Measured reactve power. D V : Voltage dstorton power. D I : Current dstorton power. P H : Harmonc actve power. D H : Harmonc dstorton power. N: Nonactve power. 4. Parameter Estmaton M H (0) P = P= P P () Q = N = Q D D D () M V I H The parameter estmaton s a procedure whch uses some samples from urements to calculate one or more unknown parameters. The used urements (samples) n the estmaton process are subject to errors, so that the estmated parameters also have assocated errors [4]. Equaton (3) shows the value of the samples obtaned by a urement devce. The ured value s close to the true value, but wth the dfference of an error. true Z Z η = (3) true where Z s the value obtaned by urement equpment; Z s the true value of the urement; and η s the random urement error that serves to model the uncertanty n the urements. The statstcal crtera used n the parameter estmaton may be quoted the maxmum lkelhood, where the goal s to maxmze the probablty that the estmated state varable s the actual value of the state varable vector [4]. Such estmator assumes that the probablty densty functon (PDF) s known. Other crtera may also be used, for example, the weghted least-squares crteron that ams to mnmze the sum of the weghted squares of the estmated ure devatons. In ths case, t s not requred that the PDF of samples or urement errors be known, but t s assumed to have Gaussan normal dstrbuton [4]. In the maxmum lkelhood crteron, t s necessary to estmate the varable x that maxmzes the lkelhood of Z [4]. Accordng to the author, the maxmum lkelhood estmate of the unknown parameter s urng thus expressed as the parameter value that mnmzes the sum of squares of the dfference between each ured value and the actual ured value, wth each squared dfference weghted by the varance of the meter error, as n (4): where: mn J x x = N = (4) Z f x σ 66

6 f x : functon used to calculate the value beng ured by the th urement. J x : urement resdual. N : number of ndependent urements. σ : varance of the th urement. Z : th urement. In one matrx approach, ths Equaton can be wrtten as n (5): T = [ ] W. D. Caetano, P. R. da Slva Jota mn J x Z f x R Z f x (5) x where: R: covarance matrx of the urements, as can be seen n (6). [ R] σ σ = σ Nm (6) 5. Ellpsod Algorthm Ths algorthm conssts n fndng a soluton to the optmzaton problem through a successon of ncreasngly smaller ellpsods, startng from the ntal ellpsod contanng the optmal pont [5]. Gven the constraned optmzaton problem descrbed n (7) [], the convergence s acheved f the objectve functon and the constrants are convex and f x R [6]. The algorthm starts wth an ellpsod E n o centered at the x pont as n (8): o { ( T ) } = arg mn * x f x qj x 0, j =,, r subject to hl x 0, l =,, s hl x 0, l =,, s (7) E x x x Q x x (8) where: q ( ) : depcts the constrants of nequalty, h( ) : represents equalty constrants descrbed as two nequalty constrants, and f ( ) : objectve functon. The ellpsod algorthm can be descrbed by recursve Equatons (9), (0) and (), whch generate a sequence of x ponts []. k Qm x = β x (9) k k k k T ( mqm k k k) ( Qm )( Qm ) T = 3 k k k k k Q β k mqm (0) k k k Q β n =,, 3 n β = n β = n where m s the gradent or sub-gradent of the functon or of the most volated constrant, accordng to the rule descrbed n () and (3) []. () gmax = max g x () 67

7 W. D. Caetano, P. R. da Slva Jota g x f gmax 0 m( ) = f x f gmax < 0 Fgure llustrates the flowchart of the ellpsod algorthm. Frstly, the varables that defne. the regon whch s contaned n the soluton (Q) (ntal ellpsod),. ntal optmal soluton (x opt ) and ts objectve functon value (f opt ), and. the algorthm parameters (for example the tolerance for calculate the gradent numercally (tol)) are ntalzed. After that, the varable receves the value of the largest constrant functon at x. Then the objectve functon value at the pont x s calculated. If r s negatve, the algorthm calculates the objectve functon gradent at pont x (f x ) and checks whether the value of f k s smaller than the optmal value acheved so far. If f k s smaller than f opt (optmal functon value), x opt and f opt receve the value of x k and f k, respectvely, otherwse the algorthm goes to the next step. If r s bgger than or equal to zero, the algorthm calculates the gradent of the most volated constrant. In the next step, the algorthm computes the varables to generate the next ellpsod based on the computed gradent vector. Ths process s repeated untl the convergence s acheved. 6. Method to Obtan the Model The data urements were performed three tmes for each load and at dfferent tmes. Durng the experments, the voltage has been changed n steps. An energy analyzer was used to collect data. Every 5 mnutes the nput voltage was ncreased by fve volts startng the urement at 0 V and endng at 30 V. The nomnal voltage s 7 V. Samples were collected every second. For each load, t was used at least 4500 urements. These data were used n the followng models. For the exponental model, the optmzaton problem s gven as n Equaton (4) [6]: T where: x= [ α P ] and f = T [ ] { α mn J x x Z f x R Z f x subject to 0 0 x s the functon descrbed n (). The same formulaton s presented for the exponental reactve power model. For ZIP model [] the parameters are estmated accordng to the optmzaton problem presented n Equaton (5) [6]: (3) (4) Fgure. Flowchart of the ellpsod algorthm. 68

8 W. D. Caetano, P. R. da Slva Jota where: T = [ ] mn J x Z f x R Z f x x Zp IP P = 0 Z p subject to 0 I p 0 Pp P s the actve power at nomnal voltage and x= P I Z P ; p p p 0 f x s the functon descrbed n (3). The same formulaton s presented for the ZIP reactve power model []. For the ZIP_mod model [], the estmated parameters must satsfy smultaneously the Equatons of actve and reactve power, so that the optmzaton problem s defned as n Equaton (6): ( α) mnαj x J x x P% I% Z% = 0 P% 0 I% 0 Z% subject to π Pθ π π Iθ π π Zθ π Sn 0 where: α = 0,5 n order to mnmze both actve and reactve power functons at the same ntensty; J the Equaton concernng the actve power wth f ( x ) gven accordng to Equaton (6); J x s x s the Equaton related to reactve power wth f ( x ) as n Equaton (7); and S n s the apparent power consumpton at nomnal voltage. The model proposes to mnmze the actve and reactve power functons smultaneously through a weghted sum, that s, both power functons are mnmzed wth the same weght. Often the objectve functons (actve and reactve power) have dfferent magntude order. Therefore, one functon can be mnmzed faster than the other. To solve ths problem, the optmal objectve functon was calculated for the actve power ( J ) and reactve ( J ) separately usng the constrants of Equaton (6), and then, the optmal values obtaned were used to calculate the objectve functon of the problem n Equaton (7): J 0 T (5) (6) J J J = λ ( λ) J (7) Ths paper uses the Ellpsod optmzaton technque for parameter estmaton obeyng the model constrants descrbed on Equatons (4), (5) and (6). 7. Methodology Knowng load mathematcal models, t s possble to obtan a model of a group of loads by smply addng the sngle models, that s, the procedure used for lnear loads (superposton theorem). In order to analyse the effect of the harmoncs n the load models and n ther aggregaton, an experment s carred out wth sx dfferent loads, and the analyss s dvded n two phases. ) Durng the frst phase, each equpment has been tested for dfferent voltage values n the range of 0.9 and.05 pu. The urements have been made repeatedly and the varables S, S, P, P, N, Q have been regstered. Usng the two groups of varables (S, P, Q and S, P, N) the mathematcal models have been obtaned by followng the procedure prevously mentoned. Consequently, two representatons (. only the fundamental fre- 69

9 W. D. Caetano, P. R. da Slva Jota quency components (S, P, Q ) and. harmoncs contrbutons (S, P, N)) have been calculated for each of the three models (Exponental, ZIP and ZIP_mod) and for each equpment. ) In the second phase, sx peces of equpment were connected smultaneously and the urements for the same varables shown n phase have been collected. Sx models have been adjusted, two representatons for the two groups of varables (. only the fundamental frequency components (S, P, Q ) and. harmoncs contrbutons (S, P, N)) for each method (Exponental, ZIP and ZIP_mod). 8. Results Followng the methodology, the experments have been done and the results are presented n ths secton. Table contans the modeled loads and ther rated power. Among the modeled loads, there are resstve loads, electroncs and motors. 8.. Phase Models Obtaned for Each Load Fgure (a) s presented to llustrate how the actve powers P and P vares when the voltage changes n steps. They are almost the same, as the actve harmonc power P H s very low. Otherwse Fgure (b) shows a bg dfference between Q M (that n fact s N) and Q. Fgure 3 presents smlar results for Lqud Crystal Dsplay urements. In ths case the dfference between actve powers P and P s more evdent. Usng these data, the models, as a functon of voltage, Exp, ZIP and ZIP_mod, are obtaned. Table presents the parameters α and β for the exponental model expressed by Equaton (7) and (8) for all modeled loads, consderng the ured data (P, N) and the fltered data (P, Q ). As the actve power suffers lttle nfluence of harmoncs, the calculated parameters (α, Po) are almost equal, even for models consderng or not harmoncs contrbuton. Otherwse, the nonactve power s strongly nfluenced by the presence of harmoncs. Table. Modeled loads. Load Compact Fluorescent Lght (6 W) Incandescent Lght Bulb (00 W) Vent-Delta Fan (45 W) LG LCD Lqud Crystal Dsplay (30 W) Personal Computer (00W) CCE Televson (Cathod Ray Tube) (75 W - MAX) Abbrevaton CFL Inc. Fan LCD PC TV (a) Fgure. Compact fluorescent urements. (a) Actve power (P, P m ) and voltage varaton; (b) Nonactve power (Q, N) and voltage varaton. (b) 70

10 W. D. Caetano, P. R. da Slva Jota (a) Fgure 3. Lqud crystal dsplay urements. (a) Actve power (P, P) and voltage varaton; (b) Nonactve power (Q, N) and voltage varaton. Table. Exponental model parameters for actve and nonactve power. Exponental model parameters (V o = 7V) (b) Load Reference parameter P Reference parameter P Reference parameter N Reference parameter Q α P o α P o β Q o β Q o CFL Inc TV Fan LCD PC Table 3 presents the parameters for actve and nonactve power Equaton consderng the ZIP model. It can be seen that there are some subtle dfferences n actve Equaton parameters, manly for the nonlnear loads lke CFL, LCD and TV. On the other hand, sgnfcant dfferences are observed for nonactve power parameters. In fact the model parameters for nonactve power are completely dfferent. For Modfed ZIP model, the results are presented n a dfferent way. As the Equatons of actve and reactve power are coupled, the results are presented n Table 4, each one for dfferent reference data S, P, Q and S, P, N respectvely. It can be seen that the results are strongly affected by the presence of harmoncs. 8.. Phase Load Aggregaton In the second phase the modeled loads, used n phase I, are turned on n the same tme, and urements of total actve and nonactve power are done. Usng these data, the models for the group of equpments data are obtaned (Collectve Model). Table 5, Table 6 and Table 7 present the obtaned models. Fnally, the model obtaned by summng up the ndvdual models s calculated (Sum of Models) by applyng superposton theorem used n Equaton (8). P Total n = P, QTotal = Q (8) = Table 8 presents a comparson among the obtaned models outputs (Collectve Model and Sum of Models), consderng or not harmoncs components (. only the fundamental frequency components (S, P, Q) and. harmoncs contrbutons (S, P, N)), for two dfferent voltage levels (0 and 30V). The lne of the ured n = 7

11 W. D. Caetano, P. R. da Slva Jota Table 3. ZIP model parameters for actve and nonactve power. ZIP model parameters (V o = 7V) Load Reference parameter P Reference parameter P P p I p Z p P o (W) P p I p Z p P o (W) CFL Inc TV Fan LCD PC Load Reference parameter N Reference parameter Q P q I q Z q Q o (Var) P q I q Z q Q o (Var) CFL Inc TV Fan LCD PC Table 4. Modfed ZIP model parameters for actve and nonactve power. ZIP_mod (V o = 7V) Load Reference parameters S, P and Q P% I% Z% Pᶿ Iᶿ Zᶿ Sn CFL Inc TV Fan LCD PC Load Reference parameters S, P and N P% I% Z% Pᶿ Iᶿ Zᶿ Sn CFL Inc TV Fan LCD PC

12 W. D. Caetano, P. R. da Slva Jota Table 5. Exponental model parameters for actve and nonactve power. Exponental model parameters (V o = 7V) Reference parameter P Reference parameter P Reference parameter N Reference parameter Q α P o α P o β Q o β Q o Table 6. ZIP model parameters for actve and nonactve power. ZIP model parameters (V o = 7V) Reference parameter P Reference parameter P P p I p Z p P o (W) P p I p Z p P o (W) Reference parameter N Reference parameter Q P q I q Z q Q o (Var) P q I q Z q Q o (Var) Table 7. Modfed ZIP model parameters for actve and nonactve power. ZIP_mod model parameters (V o = 7V) Reference parameters S, P and Q P% I% Z% Pᶿ Iᶿ Zᶿ Sn Reference parameters S, P and N P% I% Z% Pᶿ Iᶿ Zᶿ Sn Table 8. Comparson among models for two dfferent voltages and for dfferent references. V = 0V V = 30V P m or P P Q M, N Q P m P Q m or N Q Measured values Exponental model output (Equaton () and () wth the data of Table ) Collectve model a Sum of models b ZIP model output (Equaton (3) and (4) wth the data of Table 3) Collectve model a Sum of models b Modfed ZIP model output (Equaton (6) and (7) wth the data of Table 4) Collectve model a Sum of models b a These values were obtaned usng the model bult usng data for all equpment turned on smultaneously; b These values were obtaned usng the model bult usng the sum of ndvdual models. 73

13 W. D. Caetano, P. R. da Slva Jota values presents the reference data. It can be observed that the dfference between obtaned models (Exp, ZIP and ZIP_mod) of actve power for both voltage levels s almost rrelevant, consderng or not harmoncs (P m or P ). Otherwse for nonactve power the model presents very sgnfcant dfferences. For fundamental frequency component (Q ) only, the results present smlar values. Nonetheless, when the harmonc components are consdered (Q m, N), the results are completely dfferent, snce the superposton theorem cannot be appled. 9. Concluson Ths paper demonstrated the applcaton of optmzaton technque for the treatment of ellpsod constrants of three statc load models wdely used n the lterature. Ths technque was used n order to obtan real models for the loads, so that the result acheved can produce physcal meanng. All the mathematcal formulatons, as well as the methodology used to acheve the results are avalable n the ntal sectons. For most of the loads, the three models showed smlar results, except for electronc loads such as compact fluorescent lamp, montor, computer and televson. Such loads showed a reducton n power consumpton wth the ncrease n voltage level. Thus, t was evdent that among the three physcal models presented, only the ZIP_mod can fathfully represent the load, once the other models do not show curves wth negatve slope. The lterature presents these models and suggests usng the superposton theorem to obtan the model whch represents a group of loads by smple summng up the ndvdual models. Ths paper shows that the results of the obtaned models, whch do not flter the data (use only the fundamental frequency), cannot be used for load aggregaton. In other words, to apply ths knd of load model t s necessary to work wth only the fltered data of the load, otherwse the fnal model wll present very sgnfcant errors. Acknowledgements W.D. Caetano thanks CAPES (Coordnaton of mprovement of Hgher Educaton Personnel) for fnancal support. References [] Scheneder, K., Tuffner, F., Fuller, J. and Sngh, R. (00) Evaluaton of Conservaton Voltage Reducton (CVR) on a Natonal Level. Pacfc Northwest Natonal Laboratory, 4 p. [] Prce, W.W., Chang, H.-D., Clarck, H.K., Concorda, C., Lee, D.C., Hsu, J.C., Ihara, S., Kng, C.A., Ln, C.J., Mansour, Y., Srnvasan, K., Taylor, C.W. and Vaahed, E. (993) Load Representaton for Dynamc Performance Analyss. IEEE Transactons on Power Systems, 8, [3] Fellow, M.S.C., Shoults, R., Ftzer, J. and Songster, H. (99) The Effects on Reduced Voltages on the Effcency of Electrc Loads. IEEE Transactons on Power Apparatus and System, 0, [4] Kennedy, B.W. and Fletcher, R.H. (99) Conservaton Voltage Reducton (CVR) at Snohomsh Country Pub. IEEE Transacton on Power Systems, 6, [5] Hajagos, L.M. and Dana, B. (998) Laboratory Measurements and Models of Modern Loads and Ther Effect on Voltage Stablty Studes. IEEE Transactons on Power Systems, 3, [6] Neves, M.S. (0) Electrcal Power Systems and Load Modelng: A Math Model and Its Valdaton wth Case Studes. Dssertaton n Electrcal Engneerng, Federal Unversty of Juz de Fora, Juz de Fora. [7] Bokhar, A., Alkan, A., Dogan, R., Daz-Aguló, M., de León, F., Czarkowsk, D., Zabar, Z., Brenbaum, L., Noel, A. and Uosef, R.E. (04) Expermental Determnaton of the ZIP Coeffcents for Modern Resdental, Commercal, and Industral Loads. IEEE Transactons on Power Delvery, 9, [8] Wena, J.Y., Wua, Q.H., Nuttalla, K.I., Shmmna, D.W. and Cheng, S.J. (003) Constructon of Power System Load Models and Network Equvalence Usng an Evolutonary Computaton Technque. Electrcal Power and Energy Systems, 5, [9] Kumar, A. and Kumar, J. (04) ATC wth ZIP Load Model A Comprehensve Evaluaton wth Thrd Generaton FACTS n Restructured Electrcty Markets. Internatonal Journal of Electrcal Power and Energy Systems, 54, [0] Caetano, W.D., Jota, P.R.S. and Gonçalves, E.N. (03) Comparson between Statc Models of Commercal/Resdental Loads and Ther Effects on Conservaton Voltage Reducton. IEEE Internatonal Conference on Smart Energy Grd Engneerng, SEGE, Oshawa. 74

14 W. D. Caetano, P. R. da Slva Jota [] Caetano, W.D., Jota, P.R.S. and Gonçalves, E.N. (03) Modelagem de carga Estátca: Confrontação entre modelos matemátcos de cargas Resdencas/Comercas. 3th Spansh Portuguese Conference on Electrcal Engneerng (3 CHLIE), Valence. [] Takahash, R.H.C., Saldanha, R.R., Das-Flho, W. and Ramírez, J.A. (003) A New Constraned Ellpsodal Algorthm for Nonlnear Optmzaton wth Equalty Constrants. IEEE Transactons on Magnetcs, 39, [3] (00) IEEE Standard Defntons for the Measurement of Electrc Power Quanttes under Snusodal, Nonsnusodal, Balanced, or Unbalanced Condtons. IEEE Std , February. [4] Wood, A.J. and Wollenberg, B.F. (996) Power Generaton, Operaton, and Control. nd Edton, John Wley & Sons, USA, [5] Luenberger, D.G. and Ye, Y.Y. (008) Lnear and Nonlnear Programmng. 3rd Edton, Sprnger, Stanford. [6] Ecker, J.G. and Kupferschmd, M. (983) An Ellpsod Algorthm for Nonlnear Programmng. Mathematcal Programmng, 7,