A Simple Method to Estimate Power Losses in Distribution Networks

Transcription

1 A Simple Mehod o Esimae Power Losses in Disribuion Neworks Nassim A. IQTEIT, Ayşen BASA ASOY, and Bekir ÇAKI Dep. of Elecrical Engineering, Kocaeli Universiy, 8, İzmi /KOCAELİ, Turkey nassimiqei@gmail.com, aba@kocaeli.edu.r, bcakir@kocaeli.edu.r Absrac This paper presens a simple approximaed formulas o esimae acive and reacive power losses in disribuion neworks. The developed formulas are derived from Elgerd s power loss formulas considering load profiles. A simple model is also proposed o esimae he load profile based on several parameers, such as average of load demand and load limiaions. Derivaion of he load model and loss formulas are given along wih heir applicaion on a sample disribuion sysem. The resuls show ha he loss formulas depending on he proposed load profile can esimae he power losses in disribuion neworks wih high precision.. Inroducion The load profiles in convenional neworks can be esimaed from he daabase of he disribuion sysem, while in smar neworks can be obained direcly on ime from he smar sysems. Esimaed load profiles in convenional disribuion neworks helps he elecrical companies in many fields; such as, covering he energy demands of consumers, aking he economic and managemen decisions, improving he performance of neworks ec. Many researches were carried ou o deermine he model of load profiles; analyical model for deermining uncerainy in disribuion loads presened in [], depends on he calculaion of mean and variance of loading profile in each disribuion ransformer. Daily load profiles for all load ypes carried ou in [] are modeled depending on he mean, sandard deviaion and he normal disribuion of load profiles. The daily load profile in differen seasons were also sudied in []. Previous sudies on load profile need o find mean and sandard deviaion in consrucing load profile model. The proposed model of load profile in his paper consiss of a lile of load daa; ype of demands and heir average values, maximum and minimum limiaions, and he probabiliy of load disribuion beween he given limiaions. Esimaion of acive and reacive power losses in disribuion sysems is one of he main goals for any elecric uiliy company. For example, reducion of real power losses increases he performances of disribuion neworks and decreases he financial coss. Also, reducion of reacive power losses reduces he volage drops and reacive power consumpion and amelioraing loadabiliy of neworks []. For hese reasons, many equaions depending on generaed or consumed powers were proposed o calculae power losses in ransmission and disribuion sysems: Such as Kron's loss equaion [5], Branch power loss equaion [6], and Elgerd s loss equaion [7]. The equaions were used in he calculaion of power losses in he elecrical sysem, economic dispach analysis [5, 8] and deermining he opimal size and locaion of disribued generaors or capaciors in primary disribuion neworks [, 6, 9-]. These equaions have relaed wih single values of power (load, generaion or injecions power) bu hey do no have load profile funcions wih ime (i.e. daily, monhly or yearly load profiles) o calculae he average power losses in elecrical neworks. This paper presens simple approximaion power loss formulas o esimae he acive and reacive power losses, aking load profiles ino accoun. The full derivaion of he proposed power loss formulas is discussed in his manuscrip. The conens of his paper are: simple model of load profiles, acive power loss formulas based on load demands, modified power flow equaions based on power profiles, an algorihm o find acive and reacive power loss profiles, applicaion on a bus disribuion es sysem and is simulaion resuls; finally, summarizes he conribuions and conclusions of his work.. Simple Approximaed Model of Load Profiles The profiles of residenial, commercial and indusrial load demands were modeled hrough field measuremens and saisical sudy []. The daily load profiles of P () or Q () a specific probabiliy can be obained by using a normal disribuion, where he Eq. () is he general daily load profile. () =() + () () XG (): general daily load profile. μ(): he mean curve of cerain consumer. σ(): he sandard deviaion curve of a cerain consumer based on daily load curves. κ: he Gaussian disribuion value a specific probabiliy Pr(%). For example, κ=. a Pr =9% and κ =.65, Pr = 95%. The following proposed procedures presen a simple and approximaed model of daily load profile depends upon a small amoun of load daa: - According o Eq. (), if he σ()=, he daily load curve equal o he mean curve. Under his condiion, he daily load profile can be defined as XG ()= μ()= μpu () average{μ()}. A zero sandard deviaion, we can assume μpu () is he general behavior of residenial, commercial or indusrial load profile in per uni sysem. Where, () = () = - The per-uni mean profiles of residenial, commercial and indusrial were obained hrough he resuls of [], where he area under each curve is equal o one. Fig. shows he per uni mean profiles of residenial, commercial and indusrial load demands. - Approximaed load profile can be derived which are based on he values of average and minimum- maximum limis of load demands, as shown below: () () = () + () ()

2 where: () =([ ()] ) () () =() = () () = [ ()] () (5) E(Xr) = μ is he mean value of Xr, Xr is he random variable in he se {x,x,,xnd}. The subse {xmax(), μ(), xmin()} represens he maximum deviaion in he se of Xr. This subse can be used o find ou he approximae value of he λ raio. The approximae value of λ is shown in Eq. (6). () () () [ ()() ()] (6) Fig. clarifies he relaion beween maximum, minimum, and mean funcions of general load demands and heir average values. eplacing he average values of hese funcions insead of he general load funcions is he second approximaion ha can be used o simplify he calculaions of λ raio. Where he symbol is he average of x(). Eq. (7) represens he approximae value of λ, based on he range of load limis. (7) = Now Eq. () can be reformulaed based on he approximae value of λ as shown in Eq. (8). Table displays hree numerical examples and he comparison beween exac and approximaed values of λ raio, and profile values of XG. () = () + (8) Load profiles Load profiles (pu) esidenial Commercial Indusrial (a) (b) Time (s) Fig..The per uni mean load profiles based on a) he measuremen daa [], b) 5 h order polynomial funcion of curves in (a) Xr() x max pu ( ) μ pu () x ( ) min pu Δ x max pu μ pu x = min pu = mδ Fig.: Maximum, minimum, and mean funcions of he general load demand based on random subse of Xr(m). Table : Examples compare beween exac and approximae values of load profiles a specific ime. Profile of se Xr a Subse of Xr λ raio The value of profile XG ime m Exac Appx Exac Appx {.9,.75,.,.95,.9,,.97,.9,. 85,.96} {.8,.5,.7 5,.7,.,,.,.,.5,.85} {.,.,.55,.7,.9,,. 5,.86,.9,.86} {.,.9,.988} ange ±% {.,.7,.} ange ±% {.9,.5,.6} ange ±5% Power Losses. Exac Power Loss Formulas The oal acive and reacive power losses in disribuion nework wih N buses can be calculaed by Elgerd s Eq. (9) and Eq. (), respecively []. The equaions show he power losses depending on he acive and reacive power injecions. = + + (9) = + + () =[ ], =[ ], α = [α], β = [β], γ = [γ] and ξ = [ξ]. While he size of marices α, β, γ and ξ is N N. The coefficiens α, β, γ and ξ are funcions wih magniude and angle of phasor volage. Also, α and β depend on resisive enries r of impedance marix [Z], while γ and ξ depend on reacance enries x. Pi and Pj are he acive power injecions a buses i and j, respecively; Qi and Qj are he reacive power injecions a buses i and j, respecively. Where he power injecion is he difference beween he power generaions and load demands on he same bus.

3 . Power Loss Formulas Based on Load Profiles Fig. displays he change of he power injecions on he nodes of a disribuion nework a each momen of ime. The disribuion sysem was assumed perfecly balanced in he hree-phase sysem. The load profiles and he line segmens were supposed o be perfecly balanced. The line impedance of each segmen was consan wih ime, bu he sae variables (Vi () and δi ()) were no consan because hey depend on he power demands. Vi ( ) δ ( ) V i j( ) δ j( ) z = r + jx P( ), Q( ) P i ( ) Qi () i i = z θ Fig.. Load profiles on disribuion nework () P j P ( ), Q ( ).. Linearizaion Principle and Average Inegral of Power Profiles Linearizaion echnique and addiive propery of inegral were used o derive a simple power loss equaion based on power profiles. Profiles of acive and reacive load demands, power injecions and power losses are nonlinear funcions wih ime variable. Also, acive and reacive power loss relaions are nonlinear wih profiles of acive and reacive power injecions. Fig. shows a general nonlinear funcion X(), where X() could represen he power profiles of load demands (PDi (), QDi ()), power generaions (PGi (), QGi ()), power injecions (Pi (), Qi ()) a node i, or power losses (PLT (), QLT ()). Through one period T he funcion X() is divided by M regular inervals wih lengh of Δ for each one. The lengh of he regular inerval ( = ) depends on he power measuremen devices (smar, digial or convenional meers) or accuracy of analysis. X() was considered approximae linear funcion hrough each inerval m. Where Xm() is he linear approximaion of X() over inerval m. Xm_avi is he average inegral value of Xm() hrough ime Δ. is he weigh of he average inegral value of X() in inerval m. X() X m _ avi X() X m X m ( ) : Linear j Q j () Same area m- m M- M T j, m Fig.. Linearizaion principle of general nonlinear funcion X() The following relaions have been obained by using fig.. - The lengh of he regular inerval m is = and he endings of his inerval are ( ) and, where m=,,, M. - If X() is assumed a linear funcion along inerval m, hen he values of Xm_avi and can be approximaed by Eq. (). - _ = = () () () () () - The average inegral of X() hrough he period T is approximaed in Eq. () () = () = () () = () ().. Approximaed Power Loss Formulas Based on Load Profiles According o Eq. (), over inerval m he average inegral of power injecion profile (Pi (), Qi (), Pj () or Qj () ) can be represened by single poin and consan value a ime =. Dependen on approximaed values of power injecion profiles, he acive power loss profile for each inerval m was derived in he following equaion. = ( ) = ( ), ( ), ( ), ( ) =,,, = + + () Similarly, he reacive power profile for each inerval m can be calculaed by using Eq. () only by replacing γm and ξm insead of αm and βm, respecively. According o Eq. () and Eq. (), he average approximaion of acive and reacive losses based on load profiles are obained using Eq. () and Eq. (5), respecively. = + + () = + + (5) = =[ ] = =[ ] = and = = =, = =, = =, = =. While he size of marices αm, βm, γm and ξm is N N. The

4 coefficiens αm, βm, γm and ξm are calculaed for each inerval m. = = cos ( ) (6a) = = =γ = sin ( ) cos ( ) (6b) (6c) =ξ = sin ( ) (6d) =, =, =, =, =, =. Table : Load daa of a bus radial disribuion nework BUS P (pu) Q (pu) ange Probabiliy Type of (± %) (Pr%) load %+%I C I C %+5%C : residenial C : Commercial I: Indusrial. Algorihm of Power Losses Esimaion The coefficiens αm, βm, γm and ξm are calculaed for each inerval m. Power flow equaions in reference [5] were reformed o include he sae variables and power injecions a each value of m. where he sae variables Vmi and δmi can be calculaed by solving he modified power flow equaions (Eq. (7) and Eq. (8)). Numerical mehods such as Newon-aphson mehod is used o solve Eqs. (7) and (8). The power injecions Pmi and Qmi are calculaed by Eq. (). = cos ( + ) (7) P = P Di [ Di PD i Dmi DMi... P... P ] PDmi T = P Di m M Q Pmi = PGmi PDmi= PDmi Q Di= [ QD i QD i... QDmi... QDMi ] Dmi T = QDim M = sin ( + ) (8) Ym is he enry of admiance marix a values of m, i and j, Fig. 5 represens an algorihm o find he profiles and he oal average of acive and reacive power losses. m M. Applied Example The following applied example was formed hrough reconfiguring he disribuion es sysem given in [] and is load profiles were esimaed depending on he proposed simple approximaion model of load profiles. 8MVA,.5kV, bus disribuion nework, is shown in fig.6. Nodes and are unloaded. Nodes,, 8,,, and have only residenial loads. Nodes 6 and 9 have only commercial loads. Node 7 is conneced only wih indusrial load. Buses 5 and have a mixed loads. The line and he load daa are shown in able and, respecively. The base values of his nework are MVA and.5kv. Table : Line daa of a bus radial disribuion nework. BUS(n s) BUS(n r) (pu) X(pu) ½ B (pu) V,δ, P and Q Fig.5. Algorihm o find acive and reacive power losses by using simple formulas.

5 Fig.6. bus radial disribuion nework. 5. Simulaion esuls and Discussion The proposed model of load profile has been used o esimae he load profiles a each node of bus sysem. Type of load, maximum and minimum limiaions and he probabiliy percen of load disribuion were considered besides he load mean value o esimae each load profiles a each node of he sysem. The consideraion deails of our case sudy given in able, where he probabiliy percen of all disribued loads are assumed 9%. Fig.7 shows he esimaed load profiles of bus sysem a each node of he nework. Table shows he average values of acive and reacive load profiles ha shown in fig.7. The effecs of limiaions on load profiles can be noed by comparing he values of P and Q in able and. The esimaed load profiles increasing by increasing mean of load demand, a range of load limiaions, and probabiliy facor κ. M W (or) M v ar Load & P [bar] Q [bar].5 P [line] zero load Q [line] Load Load Load M W (or) M v ar Load 6 Fig.7. load profiles of a bus radial disribuion nework Table : Average values of load profile based proposed model bus P proposed model (pu) Q proposed model (pu) Load Load Load M W (or) M v ar Load Load Load Load Algorihm for power loss calculaions is achieved using MATLAB. Table 5 includes he resuls of average power losses of node radial disribuion nework. The resuls were esimaed by using he load profile and heir average and maximum values. Also, fig. 8 shows he acive and reacive power loss profiles using he proposed equaions of power losses. Table 5 confirms ha he resuls of power loss equaions are very close o he resuls of power losses when hey are direcly calculaed by power flow programs. The power loss values are differen when using single values average/maximum demand and power profiles because he power losses of he nework are a nonlinear funcion of load profiles. For example, he acive power loss based on he load profiles equals o.7pu, bu based on he average values of hese load profiles equals o.5pu. The acive power loss based on maximum values of he load profiles is equal o.66pu. The calculaion of acive power loss by using Eq. () is more accurae and effecive han using Eq. (9) and he same way calculaing reacive power loss by using Eq. (5) is more accurae and effecive han using Eq. (). Fig. 8 shows ha he maximum acive power loss is equal o.pu, while he maximum reacive loss is equal o.9pu. Fig.8 also shows ha he maximum acive and reacive power losses happened a ime :. Table 5: Power losses in bus radial disribuion nework Calculaions based on Acive/ eacive losses formula Power flow calculaion Maximum load P(pu) demands Q(pu) Average load P(pu).5.5 demands Q(pu) Load power profiles P(pu).7.7 Q(pu) Plosses & Qlosses (pu) P [bar] Q [bar] P [line] Q [line] Fig.8. Acive and reacive power losses profiles of bus radial disribuion nework Fig.9 shows he real power loss profiles a wo ype of load models: ) discree load model as shown in fig.7. ) 5 h order polynomial funcions load model. The average acive power loss in discree mode is.7pu, while in he polynomial model is.77pu. The proposed load and loss models can be used in he esimaion of power losses in neworks, economic analysis and o deermine he opimal disribuion generaors ha can be insered in disribuion neworks.

6 Power losses (pu) Ploss (Discree load model) Ploss (5h poly. load model) s Fig.9. Comparison beween he power losses profiles a discree load model and 5 h order polynomial model of load profiles. 6. Conclusions This paper presens a simple approximaed formulas o esimae he acive and reacive power losses in disribuion neworks based on load profiles. Also, i presens a simple approximaed model o esimae load demand profiles. The models of loads and losses are examined on a bus disribuion nework. The following conclusions can be drawn from his sudy: ) ype of load and is average value, he range of load changes, and he normal disribuion facor κ are he main parameers ha can effec he behavior of load profile. ) The developed loss formulas give o more precise resuls when compared o he power loss formula based on average/maximum load demand. ) he proposed formulas of acive and reacive power losses is easy o apply, is capable of using discree load profiles, and has a small error when using an appropriae number of ime inervals (M). Because of hese feaures, he simple approximaed formulas can be used no only in power loss analysis bu also economic dispach and esimae he opimal seing of disribued generaors ha can be insered ino disribuion neworks. [8] L. Wang and C. Singh, "Environmenal/economic power dispach using a fuzzified muli-objecive paricle swarm opimizaion algorihm", Elecric Power Sysems esearch, vol. 77, no., pp , 7. [9] D. Hung, N. Mihulananhan and. Bansal, "Analyical sraegies for renewable disribued generaion inegraion considering energy loss minimizaion", Applied Energy, vol. 5, pp ,. [] D. Hung, N. Mihulananhan and. Bansal, "Analyical Expressions for DG Allocaion in Primary Disribuion Neworks", IEEE Transacions on Energy Conversion, vol. 5, no., pp. 8-8,. [] D. Hung and N. Mihulananhan, "Muliple Disribued Generaor Placemen in Primary Disribuion Neworks for Loss educion", IEEE Transacions on Indusrial Elecronics, vol. 6, no., pp. 7-78,. [] M. Shahzad, I. Ahmad, W. Gawlik and P. Palensky, "Load Concenraion Facor Based Analyical Mehod for Opimal Placemen of Muliple Disribuion Generaors for Loss Minimizaion and Volage Profile Improvemen", Energies, vol. 9, no., p. 87, 6. [] W. Grady, M. Samoyj and A. Noyola, "Minimizing nework harmonic volage disorion wih an acive power line condiioner", IEEE Transacions on Power Delivery, vol. 6, no., pp , eferences [] D. Nguyen, "Modeling Load Uncerainy in Disribuion Nework Monioring", IEEE Transacions on Power Sysems, vol., no. 5, pp. -8, 5. [] J. Jardini, C. Tahan, M. Gouvea, S. Ahn and F. Figueiredo, "Daily load profiles for residenial, commercial and indusrial low volage consumers", IEEE Transacions on Power Delivery, vol. 5, no., pp. 75-8,. [] E. BOBIC, G. CATINA and G. GIGOAS, "Clusering Techniques in Load Profile Analysis for Disribuion Saions", Advances in Elecrical and Compuer Engineering, vol. 9, no., pp. 6-66, 9. [] D. Hung and N. Mihulananhan, "Loss reducion and loadabiliy enhancemen wih DG: A dual-index analyical approach", Applied Energy, vol. 5, pp. -,. [5] Saada, Power Sysems Analysis, McGraw-Hill,. [6] C. Chang, "econfiguraion and Capacior Placemen for Loss educion of Disribuion Sysems by An Colony Search Algorihm", IEEE Transacions on Power Sysems, vol., no., pp , 8. [7] O. Elgerd, Elecric Energy Sysems Theory: An Inroducion, New York: McGraw-Hill, 97.