An Efficient Method for Load Flow Solution of Radial Distribution Networks

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1 Wold Academy of Science, Engineeing and Technology An Efficient Method fo Load Flow Solution of Radial Distibution Netwoks Smaajit Ghosh, Kama Sonam Shepa Abstact This pape epots a new and accuate method fo load flow solution of adial distibution netwoks with minimum data pepaation. The node and banch numbeing need not to be sequential like othe available methods. The poposed method does not need sending node, eceiving node and banch numbes if these ae sequential. The poposed method uses the simple equation to compute the voltage magnitude and has the capability to handle composite load modelling. The poposed method uses the set of nodes of feede, lateal(s) and sub lateal(s). The effectiveness of the poposed method is compaed with othe methods using two examples. The detailed load flow esults fo diffeent kind of load modellings ae also pesented. Keywods Load flow, Feede, Lateal, Powe, Voltage, Composite, Exponential T I INTRODUCTION HE exact electical pefomance and powe flows of the system opeating unde steady state is equied in efficient way known load flow study that povides the eal and eactive powe losses of the system and voltages at diffeent nodes of the system. With the gowing maket in the pesent time, effective planning can only be assued with the help of efficient load flow study. The distibution netwok is adial in natue having high R/X atio wheeas the tansmission system is loop in natue having high X/R atio. Theefoe, the vaiables fo the load flow analysis of distibution systems ae diffeent fom that of tansmission systems. The distibution netwoks ae known as ill conditioned. The conventional Gauss Seidel (GS) and Newton Raphson (NR) method does not convege fo the distibution netwoks. A numbe of efficient load flow methods fo tansmission systems ae available in liteatue. A few methods had been epoted in liteatue fo load flow analysis of distibution systems. The analysis of distibution systems is an impotant aea of activity as distibution systems is the final link between a bulk powe system and consumes [ 3]. Smaajit Ghosh is with the Thapa Univesity, Depatment of Electical & Instumentation Engineeing, Patiala, Punjab 47004, India smaajitg@hotmail.com K.S.Shepa is with Sikkim Manipal Institute of Technology unde Sikkim Manipal Univesity, Depatment of Electical & Electonics Engineeing, Maita, Rangpo, Esat Sikkim, India. sonam_sp@yahoo.co.in The methods poposed in [4,5] wee vey time consuming and inceased the complexity. Kesting and Mendive [6] and Kesting [7] poposed a load flow technique fo solving adial distibution netwoks by updating voltages and cuents using the backwad and fowad sweeps with the help of ladde netwok theoy. Stevens et al. [8] showed that the method poposed in [6,7] became fastest but could not convege in five out of twelve cases studied. Shimohammadi et al. [9] poposed a method fo solving adial distibution netwoks with the help of diect voltage application of Kichoff s laws and pesented a banch numbeing scheme to enhance numeical pefomance of the solution method. They also extended thei method fo solving the weakly meshed distibution netwoks. Thei method needs a igoous data pepaation. Baan and Wu [0] developed the load flow solution of adial distibution netwoks by iteative solution of thee fundamental equations epesenting the eal powe, eactive powe and voltage magnitude. Renato [] poposed one method fo obtaining load flow solution of adial distibution netwoks computing the electical equivalent fo each node summing all the loads of the netwok fed though the node including losses and then stating fom the souce node, voltage of each eceiving end node was computed. Chiang [] pesented thee diffeent algoithms fo solving adial distibution netwoks based on the method of Baan and Wu [0]. Goswami and Basu [3] poposed an appoximate method fo solving adial and meshed distibution netwoks whee any node in the netwok could not be the junction of moe than thee banches i.e., one incoming and two outgoing. They had used sequential banch and node numbeing scheme. Jasmon and Lee [4] developed a load flow method fo obtaining the load flow solution of adial distibution netwoks using the thee fundamental equations epesenting the eal powe, eactive powe and voltage magnitude that had been poposed by Baan and Wu [0]. Das et al. [5] poposed a load flow method using powe convegence with the help of coding at the lateal and sub lateal nodes. Fo lage system that inceased complexity of computation. Thei method woked only fo sequential banch and node numbeing scheme. They had calculated voltage of each eceiving end node using fowad sweep. They had taken the initial guess of zeo initial powe loss. Rahaman et al. [6] poposed a method fo the impoved load flow solution of adial distibution netwoks. They had poposed a voltage equation of the ode 700

2 of fou. Ghosh and Das [7] pesented a load flow method fo solving adial distibution netwoks based on the technique with nodes beyond banches using voltage convegence. They had consideed flat voltage stat. They had shown poof of convegence and also shown that incopoation of chaging admittances educes losses and impoves voltage pofile. The main daw back of this method was that it stoes nodes beyond each banch. This method calculated cuent fo each banch by adding load cuents of nodes beyond the espective banch. Jamali et al. [8] pesented a load flow technique based on sequential banch numbeing scheme to design distibution netwok by consideing committed loads. Aavindhababu et al. [9] had shown a simple and efficient banch-to-node matix-based powe flow (BNPF) fo adial distibution systems and this method was unsuitable fo extension to optimal powe flow fo which the NR method seems to be moe appopiate. In that method any pesence of sub lateals complicates the matix fomation. Mekhame et al. [0] developed a method fo load flow solution of adial distibution netwoks using teminal conditions. Afsai et al. [] poposed a load flow method based on estimation of node voltage and assuming the loads of the nodes of lateal and thei sub lateal ae concentated at the oiginating node of the feede. They had tied to educe the computation time only. But the computation becomes vey complex when the numbe of lateals and sublateals inceases. Ranjan et al. [] poposed a new load flow technique using powe convegence chaacteistic. They had calculated voltage of each node using fowad sweep by the same voltage expession available in efeence [5]. They had calculated the total powe flow of each banch that is fed to the eceiving end node of that banch. Thei method also needed the stoage of nodes beyond each banch. They also claimed that thei algoithm could easily accommodate the composite load modeling if composition of load was known. The main disadvantage of this method was that thei method needed a epetitive seach fo connection of eceiving end node of each banch with othe nodes. In thei method, they claimed that the poposed method woked fo abitay node numbeing but emained silent egading the banch numbeing scheme. Chakaboty and Das [3] had stated that the powe convegence has the capability to handle composite load modeling. Ranjan et al. [4] had used the voltage convegence to handle the diffeent composition of load fo the same example used in efence [3]. All the poposed methods need banch numbe, sending end node and eceiving end node. The methods poposed in [3,5] needed sequential numbeing scheme. In the all the poposed methods, the examples used wee with sequential numbeing scheme. The main aim of the authos is to educe the data pepaation and to assue computation fo any type of numbeing scheme fo node and banch. If the nodes and banch numbes ae sequential, the poposed method needs only the stating node of feede, each of lateal and each of sub lateal only. The poposed method needs only the set of nodes and banch numbes of feede, each of lateals and each of sub Wold Academy of Science, Engineeing and Technology lateals only when node and banch numbes ae not sequential. The poposed method computes banch powe flow most efficiently and does not need to stoe nodes beyond each banch. The voltage of each node is calculated by using a simple algebaic equation. Although the pesent method is based on the fowad sweep, it computes efficient load flow of any complicated adial distibution netwoks vey efficiently even when banch and node numbeing scheme ae not sequential. The poposed method needs minimum data pepaation compaed to othe methods. Two examples (33 node and 69 node adial distibution netwoks) with constant powe (CP), constant cuent (CI), constant impedance (CZ), composite and exponential load modellings fo each of these examples ae consideed. The poposed method is compaed with othe existing methods [5,7,]. The initial voltage of all nodes is taken +j0 and initial powe loss of all banches ae also taken zeo. II. ASSUMPTIONS It is assumed that thee-phase adial distibution netwoks ae balanced and epesented by thei single-line diagams and chaging capacitances ae neglected at the distibution voltage levels. III. SOLUTION METHODOLOGY A single line diagam of a adial distibution netwok is shown in Fig. with sequential numbeing. In Fig., the node and banch numbeing scheme have been shown sequential. Fom Fig., set of nodes of feede, lateal and sub lateal ae FN={,,3,4,5,6}, LN={3,7,8} and SLN={7,9,0}espectively. In Fig. the set of banch numbe of feede ae FB = {,,3,4,5}, LB={6,7} and SLB = {8,9} espectively. S/S Fig. Single-line diagam of a adial distibution netwok Fig. shows when the node and banch numbeing scheme ae not sequential. Fom Fig., set of nodes of feede, lateal and sub lateal ae FN={,6,4,8,0,}, LN={4,9,3} and SLN={9,7,5} espectively. In Fig. the set of banch numbe of feede ae FB = {,7,3,9,5}, LB={6,} and SLB = {8,4} espectively. 6 x 5 6 : Banch Numbe

3 Wold Academy of Science, Engineeing and Technology S/S Fom Fig. and Fig., the sub lateal has two banches, the lateal has two banches and the feede has five banches. Let the feede is denoted by, lateal by and sub lateal by 3 in Fig. and Fig.. Hee the two dimensional aay FN denotes the node of feede, each lateal and each sub lateal whee the fist numbe of the aay indicates feede, lateal and sub lateal. At fist feede is kept, then lateal and sub lateal. The second numbe denotes the ode of the node of the set. Fom Fig., the nodes of feede, lateal and sub lateal ae shown below. FN(,) =, FN(,) =, FN(,3) = 3, FN(,4) = 4, FN(,5) = 5 and FN(,6) = 6 FN(,) = 3, FN(,) = 7 and F(,3) = 8 and FN(3,) = 7, FN(3,) = 9 and F(3,3) = 0. Fom Fig., the banches of feede, lateal and sub lateal ae shown below. FB(,) =, FB(,) =, FB(,3) = 3, FB(,4) = 4 and FB(,5) = 5 FB(,) = 6 and FB(,) = 7 and FB(3,) = 8 and FB(3,) = 9. Fom Fig., the nodes of feede, lateal and sub lateal ae shown below. FN(,) =, FN(,) = 6, FN(,3) = 4, FN(,4) = 8, FN(,5) = 0 and FN(,6) = FN(,) = 4, FN(,) = 9 and F(,3) = 3 and FN(3,) = 9, FN(3,) = 7 and F(3,3) = 5. Fom Fig., the banches of feede, lateal and sub lateal shown below. FB(,) =, FB(,) = 7, FB(,3) = 3, FB(,4) = 9 and FB(,5) = 5 FB(,) = 6 and FB(,) = and FB(3,) = 8 and FB(3,) = 4. Let jj = FB(i,j), m = FN(i,j+) and m = FN(i,j). We have V(m) = V(m) I(jj)Z(jj) () Let V(m) = V(m) δ V(m) = V(m) δ Z(jj) = Z(jj) ϕ = R(jj) + jx(jj) and I(jj) = I (jj) θ Voltage of node m is expessed by Fig. without sequential numbeing scheme 6 0 x : Banch Numbe 5 V(m) s + s P (jj) Q (jj) Z(jj) = V(m) () V(m) whee P s (jj) and Q s (jj) ae the eal and eactive powes coming out fom the node m. The detailed deivation has been shown in Appendix A. Voltage of node m can also be calculated using the following expession also: V(m) ± V(m) 4 {P (jj) + Q (jj)} Z(jj) V(m) = (3) whee P (jj) = P s (jj) LP(jj) and Q (jj) = Q s (jj) LQ(jj) ae the eal and eactive powe fed though the node m. Equation () is used to calculate V(m) due to its simplicity. The cuent though the banch jj is expessed by V(m) V(m) I(jj) = (4) Z(jj) The eal and eactive powe loss of banch jj is expessed by LP(jj)= I(jj) R(jj) (5) and LQ(jj)= I(jj) X(jj) (6) P s (jj) = Sum of eal powe load of all nodes afte the banch jj plus the eal powe loss of all the banches afte the banch jj including the banch jj also. Q s (jj)= Sum of eactive powe load of all nodes afte the banch jj plus the eactive powe loss of all the banches afte the banch jj including the banch jj also. To discuss the calculation of P s (jj) and Q s (jj), P s (jj) and Q s (jj) fo sub lateal(s), lateal(s) and feede ae calculated at fist with an assumption that they ae sepaated. Fo the sub lateal: P s [FB(3,)] = PL[FN(3,3)] + LP[FB(3,)] (7) P s [FB(3,)] = PL[FN(3,)] + LP[FB(3,)] + P s [FB(3,)] P s [FB(,)] = PL[FN(,3)] + LP[FB(,)] P s [FB(,)] = PL[FN(,)] + LP[FB(,)] + P s [FB(,)] P s [FB(,5)] = PL[FN(,6)] + LP[FB(,5)] P s [FB(,4)] = PL[FN(,5)] + LP[FB(,4)] + P s [FB(,5)] P s [FB(,3)] = PL[FN(,4)] + LP[FB(,3)] + P s [FB(,4)] P s [FB(,)] = PL[FN(,3)] + LP[FB(,)] + P s [FB(,3)] P s [FB(,)] = PL[FN(,)] + LP[FB(,)] + P s [FB(,)] Fom (7), (8) and (9), we can conclude the following: Fo the end banch P s [FB(i,j)] = PL[FN(i,j+)] + LP[FB(i,j)] (0) and fo othe banches, P s [FB(i,j)] =PL[FN(i,j+)]+LP[FB(i,j)]+P s [FB(i,j+)] () Equations(0) and () shows genealized expessions fo the computation of P s s though the feede, lateal and sub lateal when they ae sepaated. Similaly, the following ae the genealized expessions fo Q s s: (8) (9) 70

4 Fo the end banch Q s [FB(i,j)] = QL[FN(i,j+)] + LQ[FB(i,j)] () and fo othe banches, Q s [FB(i,j)]=QL[FN(i,j+)]+LQ[FB(i,j)]+Q s [FB(i,j+)] (3) Now fom Fig. and Fig., we have the following: Sub lateal is connected to lateal at the node F(,). Theefoe, powe flow though the banch FB(,) becomes P s [FB(,)] = PL[FN(,)] + LP[FB(,)] + P s [FB(,)] + P s [FB(3,)] (4) and Q s [FB(,)] = QL[FN(,)] + LQ[FB(,)] + Q s [FB(,)] + Q s [FB(3,)] (5) The lateal is connected to feede at the node F(,3). Theefoe, powe flow though the banch FB(,) becomes P s [FB(,)] = PL[FN(,3)] + LP[FB(,)] + P s [FB(,3)] + P s [FB(,)] (6) and Q s [FB(,)] = QL[FN(,3)] + LQ[FB(,)] + Q s [FB(,3)] + Q s [FB(,)] (7) Fom the above discussion, it can be concluded that the common nodes of among the sub lateal(s) and lateal(s) as well as that of feede and lateal(s) must be maked at fist. If FN(i,j) be the node of lateal which is the souce node of the sub lateal also o be the node of feede which is the souce node of the lateal also, the banch numbe FB(i,j ) is equied to be stoed. The poposed logic checks the common nodes of lateal(s) and sub lateal(s) [ fist node of the sub lateal(s)] and also stoes the banch numbe. If the node FN(i,j) of the lateal and fist node FN(x,) of the sub lateal ae identical, the banch FB(i,j ) of the lateal to be stoed in the memoy say the vaiable mm[tn ] whee TN is the total numbe denoting the sum of numbes of feede, lateal(s) and sub lateal(s) and the sub lateal numbe is also stoed in the aay mn[tn ]. Hee TN shows the total memoy size of the aay. Similaly, the common nodes of lateal(s) and feede ae found out and the banch numbe of the feede coesponding to the common node of feede and lateal ae stoed in mm[tn ] and simultaneously lateal numbe is stoed in mn[tn ]. The banches of lateal(s) and feede(s) ae checked with the banches stoed in the aay mm[tn ]. If any banch numbe of lateal and feede matches with any element of mm[tn ], say the banch numbe of FB(i,j) matched with mm[], the P s and Q s fo the banch FB(i,j) will be P s [FB(i,j)] = PL[FN(i,j+)] + LP[FB(i,j)] + P s [FB(i,j+)] + P s [FB(mn[],)] (8) and Q s [FB(i,j)] = QL[FN(i,j+)] + LQ[FB(i,j)] + Q s [FB(i,j+)] + Q s [FB(mn[],)] (9) whee mn[] is the numbe of lateal o sub lateal depending of the value of i. Fom above discussion, it is clea that the poposed method does not depend upon the node and banch numbeing. To make the computation of P s and Q s faste, the logic used in the poposed method is descibed below: Step : Get the numbe of Feede(A), lateal(s) (B) and sub lateal(s) (C). Step : TN = A + B + C Step 3 : Read total numbe of nodes of feede, each Wold Academy of Science, Engineeing and Technology lateal and sub lateal espectively i.e., N(i) fo i =,,..,TN. Step 4 : Get the status of numbeing scheme. Step 5 : If it is sequential, ask fo the stating node of feede, each lateal and sub lateal espectively. Go to Step 7. Step 6 : If it is not sequential, ead the set of nodes as well as banches of feede, each lateal and sub lateal espectively. Step 7 : Find the common nodes of sub lateal(s) and lateal(s) i.e., FN(i,) fo i = TN to TN C+ fom FN(i,j) fo j =,,,N(i) and i = TN C to TN C B. Stoe them in mm(i) fo i =,,..,C and stoe the banch of lateal FB(i,j ) coesponding to the node FN(i,j) in mn(i) fo i=,,..,c. Step 8 : Find the common nodes of lateal(s) and Feede i.e., FN(i,) fo i = TN C to TN C B+ fom FN(,j) fo j =,,,N(). Stoe them in mm(i) fo i = C+,..,C+B and the banch of feede FB(i,j ) coesponding to the node FN(i,j) in mn(i) fo i = C+,..,C+B. Step 9 : Calculate P s [FB(i,j)] and Q s [FB(i,j)] fo j = N(i),,, and fo i = TN to TN C+ using (0) o () and () o (3) espectively. Step 0 Step : Calculate P s [FB(i,j)] and Q s [FB(i,j)] fo j = N(i),,, and fo i = TN C to TN C B+ using (8) and (9) espectively with a check of FB(i,j) fo j = N(i),,, and fo i = TN C to TN C B+ with mn(k) fo k =,,..,C. : Calculate P s [FB(,j)] and Q s [FB(,j)] fo j = N(i),,, using (8) and (9) espectively with a check of FB(,j) fo j = N(i),,, with mn(k) fo k =C+,,C+B. IV. LOAD MODELLING A balanced load that can be epesented eithe as constant powe, constant cuent, constant impedance o as an exponential load is consideed hee. The geneal expession of load is shown below. P(m) = P n [a 0 + a V(m) + a V (m) + a 3 V e (m)] (0) Q(m)= Q n [b 0 + b V(m) + b V (m) + b 3 V e (m)] () whee, P n and Q n ae nominal eal and eactive powe espectively and V(m) is the voltage at node m. Fo all the loads, ( 0) and () ae modeled as a 0 + a + a + a 3 =.0 () b 0 + b + b + b 3 =.0 (3) Fo constant powe (CP) load a 0 = b 0 = and a i = b i = 0 fo i =,, 3. Fo constant cuent (CI) load a = b = and a i = b i = 0 fo i = 0,, 3. Fo constant impedance (CZ) load a = b = and a i = b i = 0 fo i = 0,, 3. Composite load modelling

5 is combination of CP, CI and CZ. Fo exponential load a 3 = b 3 = and a i = b i = 0 fo i = 0,, and e and e ae.38 and 3. espectively [3]. V. ALGORITHM FOR COMPUTATION OF LOAD FLOW To calculate the node voltages and banch cuents and the total system loss, a initial guess of zeo eal and eactive powe loss is assumed. Also flat voltage stat is used. The convegence citeia is such that if Max V old [FN(i,j)] V New [FN(i,j)] < ε, fo i =,,..,TN and j =,,.,N(i)=total numbe of nodes of FN(i). The following ae the steps fo load flow calculation: Step : Get the numbe of Feede(A), lateal(s) (B) and sub lateal(s) (C). Step : TN = A + B + C Step 3 : Read the total numbe of nodes N(i) of feede, lateal(s) and sub lateal(s) fo i =,,,TN Step 4 : Read the nodes and banch numbes of feede, lateal(s) and sub lateal(s) i.e., FN(i,j) fo j =,,,N(i) and i =,,.,TN if these ae not sequential.. Step 5 : Read eal and eactive powe load at each node i.e., PL[FN(i,j)] and QL[FN(i,j)] fo j =,3,..,N(j) and i =,,..,TN. Step 6 : Initialize PL[FN(,)] = 0.0 and QL[FN(,)] = 0.0 Step 7 : Read the banches of feede, lateal(s) and sub lateal(s) i.e., FB(i,j) fo j =,,,N(i) and i =,,.,TN. Step 8 : Read esistance and eactance of each banch i.e., R[FB(i,j)] and X[FB(i,j)] fo j =,3,..,N(j) and I =,,..,TN. Step 9 : Read base kv and base MVA, Total numbe of iteation (ITMAX), ε (0.0000) Step 0 : Compute the pe unit values of PL[FN(i,j)] and QL[FN(i,j)] fo j =,3,..,N(j) and i =,,..,TN as well as R[FB(i,j)] and X[FB(i,j)] fo j =,,3,..,N(j) and i =,,..,TN. Step : Set PL[FN(i,j)] = PL[FN(i,j)] and QL[FN(i,j)] = QL[FN(i,j)] fo j =,3,..,N(j) and i =,,..,TN Step : Set LP[FB(i,j)] = 0.0 and LQ[FB(i,j)] = 0.0 fo all j =,,,N(i) and i =,,.,TN. Step 3 : Set V[FN(i,j)] =.0 + j0.0 fo j =,,,N(i) and i =,,.,TN and set V[FN(i,j)] = V[FN(I,j)] fo j =,,,N(i) and i =,,.,TN. Step 4 : Use the Step7 to Step (At 3.0) to calculate the banch cuents of each feede, lateal(s) and sub lateal(s) espectively. Step 5 : Set IT = Step 6 : Set PL[FN(i,j)] = PL[FN(i,j)] and QL[FN(i,j)] = QL[FN(i,j)] fo j =,3,..,N(j) and i =,,..,TN Step 7 : Use pope load modeling using (0) and (). Step 8 : Compute voltage V[FN(I,j)] using () fo j = Wold Academy of Science, Engineeing and Technology ,3,..,N(j) and i =,,..,TN. Step 9 : Compute ΔV[FN(i,j)] = V[FN(i,j)] V[FN(i,j)] fo j =,3,..,N(j) and i =,,..,TN. Step 0 : Compute cuent I[FB(i,j)] using (4) fo j =,,3,..,N(j) and i =,,..,TN. Step : Set V[FN(i,j)] = V[FN(i,j)] fo j =,,3,..,N(j) and I =,,..,TN. Step : Compute LP[FB(i,j)] and LQ[FB(i,j)] fo all j =,,,N(i) and i =,,.,TN using (5) and (6) espectively. Step : Find ΔV max fom ΔV[FN(i,j)] fo j =,3,..,N(j) and i =,,..,TN. Step 3 : If ΔV min go to Step 6 else go to Step 4. Step 4 : IT = IT + Step 5 : If IT ITMAX go to Step 6 else wite NOT CONVERGED and go to Step 7. Step 6 : Wite SOLUTION HAS CONVERGED and display the esults : Total Real and Reactive Powe Losses, Voltages of each node, minimum value of voltage and its node numbe and total eal and eactive powe load fo CP, CI, CZ, Composite and Exponential Load Modelling. Step 7 : Stop VI. EXAMPLES To demonstate the effectiveness of the poposed method, the following two examples ae consideed hee: The fist example is 33 node adial distibution netwok (nodes have been enumbeed with Substation as node ) shown in Fig. 3. Data fo this system ae available in [5]. Real and eactive powe loss fo CP, CI, CZ, Composite and Exponential load modeling as well as the minimum voltage and its node numbe is shown in Table. Base values fo this system ae.66 kv and 00 MVA espectively. S/S Fig Node Radial Distibution Netwok [5]

6 S/S The second example is 69 node adial distibution netwok (nodes have been enumbeed with Substation as node ). Data fo this system ae available in [0]. Real and eactive powe loss fo CP, CI, CZ, Composite and Exponential load modeling as well as the minimum voltage and its node numbe is shown in Table. Base values fo this system ae.66 kv and 00 MVA espectively Fig node adial distibution netwok[0] Wold Academy of Science, Engineeing and Technology node adial distibut ion netwok [0] CP V 65 = CI V 65 = CZ V 65 = Com posit e Expo nenti al V 65 = V 65 = The compaison of elative CPU Time of the poposed method with the othe existing methods [5,7,] fo constant powe load modelling has been shown in Table II. All simulation woks have been caied out in Celeon Pocesso GHz. TABLE II COMPARISON OF RELATIVE CPU TIMEOF THE PROPOSED METHOD WITH OTHER EXISTING METHODS [5,7,] FOR CONSTANT POWER LOAD MODELING Methods Examples VII CONCLUSION Example 3 Example 4 CPU Time CPU Time Poposed method D.Das et al. [5].90.3 S.Ghosh and D.Das [7].4.8 Ranjan and D.Das [] In all cases Composite Load = 40%CP + 30%CI + 30% CZ has been consideed. Compaison of CPU time of the poposed method with the methods [5,7,] is shown in Table. TABLE I REAL POWER LOSS, REACTIVE POWER LOSS, MINIMUM VOLTAGE FOR CP, CI, CZ, COMPOSITE AND EXPONENTIAL LOAD MODELLING FOR 33 NODE AND 69 NODE RESPECTIVELY Minimu m Voltage 33 node adial distibut ion netwok [5] Type of Load Real (kw) Total Load Reactive (kva) Real (kw) Powe Loss Reactiv e (kva) Minimum Voltage (p.u.) CP V 8 = CI V 8 = CZ V 8 = Com posit e Expo nenti al V 8 = V 8 = An efficient method fo load flow solution of adial distibution netwok has been poposed in this pape. The poposed method educes the data pepaation. The poposed method simply needs stating nodes of feede, lateal(s) and sub lateal(s) and no data of banch numbes fo sequential numbeing scheme. If the node and banch numbes ae not sequential, only node numbes and banch numbes of each feede, lateal(s) and sub lateal(s) ae equied. Theefoe, the poposed method consumes less compute memoy. The poposed method uses the simple voltage equation. The poposed method takes the zeo initial loss fo computation of voltage of each node and consides flat voltage stat to incopoate voltage convegence. The poposed method ovecomes the shotfalls of the methods epoted in [5,7,]. Effectiveness of the poposed method has been demonstated by two examples (33 node and 69 node adial distibution netwoks) with constant powe load, constant cuent load, constant impedance load, composite load and exponential load fo each of these examples. The efficiency of the poposed method in tems of CPU time has been checked by compaing it with the othe existing methods [5,7,]. 705

7 Fom the poposed voltage equation a suitable stability index can also be fomed. APPENDIX Let FB(i,j) = jj, FN(i,j) = m and FN(i,j+) = m. Theefoe, we have V(m) = V(m) I(jj)Z(jj) (A) Let V(m) = V(m) δ V(m) = V(m) δ Z(jj) = Z(jj) ϕ = R(jj) + jx(jj) and I(jj) = I (jj) θ Theefoe, (A) becomes V(m) (cosδ + jsinδ ) i.e., V(m) cosδ = V(m) (cosδ+ jsinδ ) I (cosθ + jsinθ ) {R(jj) + jx(jj)} + jv(m)sinδ = V(m) cosδ I(jj) {R(jj)cosθ + X(jj)sinθ } +j[ V(m) sinδ I(jj) {(X(jj)cosθ R(jj)sinθ )} V(m) cosδ = V(m) cosδ I(jj) {R(jj)cosθ + X(jj)sinθ } V(m) sinδ and = V(m) sinδ I(jj) {X(jj)cosθ R(jj)sinθ } Fom (A) and (A3), we have (A) (A3) V(m) = V(m) V(m) I(jj) cosδ {R(jj)cosθ + X(jj)sinθ } + I(jj) {R (jj) + X (jj)} V(m) I(jj) sin δ {X(jj)cos θ R(jj)sin θ } = V(m) V(m) I(jj) {R(jj)(cosδcosθ sinδsinθ ) + X(jj)(cosδsinθ + sinδcosθ )} + I(jj) { R (jj) + X (jj) } = V(m) V(m) I(jj) {R(jj)(cos(δ+ θ) + X(jj)sin(δ+ θ)} + I(jj) Z(jj) = V(m) ϕ V(m) I(jj) Z(jj) {cos cos(δ+ θ) + sin sin(δ+ θ)} + I(jj) Z(jj) = V(m) V(m) I(jj) Z(jj) cos( ϕ δ θ) + I(jj) Z(jj) Since, φ δ θ is vey vey small and hence cos( φ δ θ) Theefoe, V(m) = V(m) V(m) I(jj) Z(jj) + I(jj) Z(jj) Wold Academy of Science, Engineeing and Technology ϕ i.e., V(m) = V(m) I(jj) Z(jj) i.e., V(m) = V(m) I(jj) Z(jj) (A4) Again, and also + Q (jj) P (jj) I(jj) = (A5) V(m) s + Q s(jj) P (jj) I(jj) = (A6) V(m) whee P (jj) = P s (jj) LP(jj) and Q (jj) = Q s (jj) LQ(jj) ae the eal and eactive powe fed though the node m. Using (A6), (A4) can be witten as P s (jj) + Q s(jj) Z(jj) V(m) = V(m) (A7) V(m) Using (A5), (A4) can be witten as i.e., i.e., i.e., V(m) + P (jj) Q (jj) Z(jj) = V(m) (A8) V(m) = + V(m) V(m) V(m) P (jj) Q (jj) Z(jj) V(m) V(m) V(m) + P (jj) + Q (jj) Z(jj) = 0 V(m) V(m) ± V(m) 4 {P (jj) + Q (jj)} Z(jj) = (A9) REFERENCES [] N. Vempati, R.R.Shoults, M.S. Chen, L. Schwobel, Simplified feede modeling fo load flow calculations, IEEE Tans. on Powe Systems;vol.,no.,pp.68 74,987. [] T.H.Chen, M.Chen, K.J. Hwang, P. Kotas, E.A.Chebli, Distibution system powe flow analysis a igid appoach, IEEE Tans. on Powe Delivey; vol.6,no.3 pp. 46 5,98. [3] Distibution automation: a pactical tool fo shaping a moe pofitable futue: Special epot, Electical Wold,pp , Decembe 986. [4] S.Iwamoto, Y.A.Tamua, Load flow calculation method fo ill conditioned powe systems, IEEE Tansactions Powe Appaatus and Systems, vol. PAS 00, No. 4, pp ,98. [5] D. Rajicic, Y. Tamua, A modification to fast decoupled load flow fo netwoks with high R/X atios, IEEE Tansactionson Powe Systems; vol. 3, no., pp ,988. [6] W.H.Kesting, D.L. Mendive, An Application of Ladde Theoy to the Solution of Thee-Phase Radial Load-Flow Poblem, IEEE Tansactions on Powe Appaatus and Systems; vol. PAS 98 no.7, pp , 976. [7] W.H.Kesting, A Method to Teach the Design and Opeation of a Distibution System, IEEE Tansactions on Powe Appaatus and Systems; vol. PAS 03, no.7, pp , 984. [8] R.A. Stevens, et al., Pefomance of Conventional Powe Flow Routines fo Real Time Distibution Automation Application, Poceedings 8 th Southeasten Symposium on Systems Theoy: IEEE Compute Society: 96 00, 986. [9] D, Shimohammadi, H.W. Hong, A. Semlyn, G.X. Luo, A Compensation Based Powe Flow Method fo Weakly Meshed Distibution and Tansmission Netwok, IEEE Tansactions on Powe Systems, vol. 3, no., pp ,

8 [0] M.E. Baan ME and F.F. Wu, Optimal Sizing of Capacitos Placed on a Radial Distibution System, IEEE Tansactions on Powe Delivey; vol. 4, no., pp , 989. [] C.G. Renato, New Method fo the Analysis of Distibution Netwoks, IEEE Tansactions on Powe Delivey; vol. 5, no., pp. 9 3, 989. [] H.D.Chiang, A Decoupled Load Flow Method fo Distibution Powe Netwoks: Algoithms Analysis and Convegence Study, Intenational Jounal of Electical Powe Systems; vol. 3, no.3, pp.30-38,99. [3] S.K. Goswami and S.K. Basu, Diect Solutions of Distibution Systems, IEE Pat C (GTD), vol.38, no., pp.78 88,99. [4] G.B.Jasmon and L.H.C.C. Lee, Stability of Load-Flow Techniques fo Distibution System Voltage Stability Analysis, Poceedings IEE Pat C (GTD), vol.38, no. 6, pp , 99. [5] D. Das, H.S.Nagi and D.P. Kothai, Novel Method fo solving adial distibution netwoks, Poceedings IEE Pat C (GTD), vol.4, no. 4, pp. 9 98, 99. [6] T.K.A. Rahman and G.B. Jasmon, A new technique fo voltage stability analysis in a powe system and impoved loadflow algoithm fo distibution netwok, Enegy Management and Powe Delivey Poceedings of EMPD '95; vol., pp.74 79, 995. [7] S. Ghosh and D. Das, Method fo Load Flow Solution of Radial Distibution Netwoks, Poceedings IEE Pat C (GTD), vol.46, no.6,pp , 999. [8] S. Jamali. M.R.Javdan. H. Shatei and M. Ghobani, Load Flow Method fo Distibution Netwok Design by Consideing Committed Loads, Univesities Powe Engineeing Confeence, vol.4, no.3, pp , sept.006. [9] P. Aavindhababu, S. Ganapathy and K.R. Naya, A novel technique fo the analysis of adial distibution systems, Intenational Jounal of Electic Powe and Enegy Systems, vol. 3, pp. 67 7, 00. [0] S.F. Mekhame et.al., Load Flow Solution of Distibution Feedes: A new contibution, Intenational Jounal of Electic Powe Components and Systems,vol. 4, pp , 00. [] A. Afsai, S.P. Singh, G.S. Raju, G.K.Rao, A fast powe flow solution of adial distibution netwoks, Intenational Jounal Electic Components and Systems, vol. 30, pp ,00. Wold Academy of Science, Engineeing and Technology [] R. Ranjan and, D. Das, Simple and Efficient Compute Algoithm to Solve Radial Distibution Netwoks, Intenational Jounal of Electic Powe Components and Systems, vol.3, no.,: pp.95 07, 003. [3] M.Chakavoty and D.Das, Voltage stability analysis of adial distibution netwoks, Intenational Jounal of Electic Powe and Enegy Systems, vol.3, pp. 9 35, 00. [4] R.Ranjan, B.Venkatesh and D.Das, Voltage Stability Analysis of Radial Distibution Netwoks, Intenational Jounal of Electic Powe Components and Systems, vol. 33, pp.50 5, 004. [5] M.E.Baan and F.F. Wu, Netwok Reconfiguation in Distibution Systems fo Loss Reduction and Load Balancing, IEEE Tansactions on Powe Delivey; vol.4, no., pp , 989. Smaajit Ghosh seving as a Pofesso in Thapa Univesity, Depatment of Electical and Instumentation Engineeing. He did his B.Tech., M.Tech. in Electical Machines and Powe Systems fom Calcutta Univesity in 994 and 996 espectively. Finally, he did his Ph.D. fom Indian Institute of Technology, Khapagpu, India in 000. His eseach aeas include load flow study, netwok econfiguations, optimum capacito allocation, application of soft computing in Elctical Powe Distibution Systems. Kama Sonam Shepa seving as a Reade in Sikkim Manipal Institute of Technology, Depatment of Electical and Electonics Engineeing unde Sikkim Manipal Univesity. He did his B.E. fom Malviya Regional Engineeing College, Jaipu in Electical Engineeing and M.Tech. in Powe Electonics and Dives fom Indian Institute of Technology, Khaagpu, India. Cuently he is doing Ph.D. in Sikkim Manipal Univesity. 707