Computational Analysis of IEEE 57 Bus System Using N-R Method

Transcription

1 Vol 4, Issue 11, November 2015 Computatioal Aalysis of IEEE 57 Bus System Usig N-R Method Pooja Sharma 1 ad Navdeep Batish 2 1 MTech Studet, Dept of EE, Sri SAI Istitute of Egieer &Techology, Pathakot, Idia 2 Assistat professor, Dept of EE, Sri SAI Istitute of Egieer &Techology, Pathakot, Idia ABSTRACT: I this research work, the power flow problem, also called as the load flow problem, has bee dealt with The load flow solutio gives the complex voltages at all the buses ad the complex power flows i the lies To obtai power flow solutio, the most popular Newto-Raphso method is used The method has bee used to obtai power flow solutios ad is tested o IEEE 57- bus distributio system I IEEE 57-bus system,, the total power geerated were 1278MW whereas the power demad were 1250MW thus a loss of 28 MW ad the optimal cost rages from 4213$/MVA-hr to 4683$/MVA-hr KEYWORDS: Power Aalysis, Bus, Computatio, MATPOWER IINTRODUCTION Load flow aalysis has the sigificat importace i the study of power systems power Power or load flow study deals with the study of various quatities of the power systems such as real power, reactive power, ad magitude of voltage ad agle of voltage Load flow study is doe o a power system to esure that geeratio supplies load ad losses From load flow study, we ca esure that bus voltage should be ear to the rated values ad the geeratio operates withi real ad reactive power limits We ca isure that trasmissio lies ad trasformers are ot overloaded The objective of load flow aalysis is i the plaig stage of ew etworks, addig ad erectio of a ew etwork to the existig substatio It gives the odal voltages ad phase agles, power ijectio at all t the buses ad power flows through itercoectig power chaels It is helpful i determiig the best locatio as well as optimal capacity of proposed geeratig statio, substatio ad ew lies It determies the voltage of the buses ad keeps withi the closed toleraces From the load flow aalysis the locatio of maximum voltage variatio ca be obtaied Due to load fluctuatios durig peak load coditios uder voltage problems also will be preset But durig some other load coditios over voltage or over load coditios may occur May types of software are available for load flow aalysis of huge power system [1] David I Su et al (1984) proposed a classical optimal power flow problem with a o separable objective fuctio ca be solved by a explicit Newto approach With this approach efficiet, robust solutios ca be obtaied for problems of ay practical size or kid [2] Paulo A N et al (2000) preseted a paper dealig with the formatio of sparse matrix formulatio for the solutio of ubalaced three-phase power systems usig the Newto Raphso method The threephase curret ijectio equatios are writte i rectagular coordiates resultig i a order 6 system of equatios [3] Ambriz-Perez (2002) preseted advaced load flow models for the static VAr compesator (SVC) The models takes ito accout the existig load flow (LF) ad optimal power flow (OPF) Newto algorithms I this paper focus is laid o the ew models depart from the geerator represetatio of the SVC ad are based istead o the variable shut susceptace cocept [4] For carryig put load flow aalysis, umbers of methods are available but the procedure ivolved is same for all ad the particular steps ivolved i load flow aalysis are preseted as: Copyright to IJAREEIE DOI: /IJAREEIE

2 Vol 4, Issue 11, November 2015 Fig 1 Steps for load flow aalysis Newto Raphso method is the best opted method for solvig o-liear load flow equatios [5] The umber of iteratios ivolved i Newto Raphso method is idepedet of umber of buses cosidered, hece power flow equatios ca be solved just i few iteratios Keepig i view all these advatages, Newto Raphso method is popularly used for load flow studies i a power system [6] I usig Newto Raphso method, a direct solver is used to solve the liear systems Basically a iterative techique is used i this method for obtaiig optimal power flow solutio Mathematical model for NR method I applicatio of the NR method, we have to first brig the equatios to be solved, to the form f(x 1, x 2 ) =0 where x 1, x2 x are the ukow variables to be determied Let us assume that the power system has 1 PV buses ad 2 PQ buses I polar coordiates the ukow variables to be determied are: (i) θ i, the agle of the complex bus voltage at bus i, at all the PV ad PQ buses This gives us ukow variables to be determied (ii) V i, the voltage magitude of bus i, at all the PQ buses This gives us 2 ukow variables to be determied Therefore the total umber of variables to be computed is , for which cosistet equatios eeds to be solved The equatios are as uder; P i = P i,spec P i,cal = 0 Q i = Q i,spec Q i,cal = 0 Where P i,spec = Specified active power at bus i Q i,spec = Specified reactive power at bus i P i, cal = Calculated value of active power usig voltage estimates Q i, cal = Calculated value of reactive power usig voltage estimates ΔP = Active power residue ΔQ = Reactive power residue Usig polar coordiate system, the voltage magitude equatio, the real ad reactive power equatios ca be expressed as: V i = V i (cos θ + j si θ) P i = V i Q i = V i j=1 j=1 V j (G ij cos θ ij + B ij si θ ij ) V j (G ij cos θ ij G ij si θ ij ) Whereθ ij = θ i θ j, which is the agle differece betwee buses i ad bus j Copyright to IJAREEIE DOI: /IJAREEIE

3 Vol 4, Issue 11, November 2015 The power flow equatios ca be expaded ito Taylor series usig Newto Raphso method as follows: θ P Q = J V V P Q = H N K L P Q = J 1 J 2 J 3 J 4 θ V D 1 V Where P = P 1 P 2 Ad Q = θ = P 1 Q 1 Q 2 Q m θ 1 θ 2 θ 1 V 1 V 2 V = V m V 1 V D = V 2 V m H is a ( 1) ( 1) matrix, ad its elemet ish ij = P i θ j N is a ( 1) m matrix, ad its elemet is N ij = V j P i V j K is a m ( 1) matrix, ad its elemet is K ij = Q i θ j L is a m m matrix, ad its elemet is L ij = V j Q i V j These parameters are the defiig oe i formig Jacobia matrix ad hece to perform load flow solutio Calculatio of P cal ad Q cal : The real ad reactive powers ca be calculated usig the followig equatios; P i,cal = P i = j =1 V i V j (G ij cos θ ij + B ij si θ ij ) P i = G ii V i 2 + V i j =1 V j (G ij cos θ ij + B ij si θ ij ) Copyright to IJAREEIE DOI: /IJAREEIE

4 Vol 4, Issue 11, November 2015 Q i,cal = Q i = j =1 Q i = B ii V i 2 + V i V j j =1 V i V j (G ij si θ ij B ij cos θ ij ) (G ij si θ ij B ij cos θ ij ) The powers are computed at ay (r+1) th iteratio by usig the voltages available from previous iteratio The elemets of the Jacobia are foud usig the above equatios as: IEEE 57 bus system The stadard IEEE 57-bus system cosists of 80 trasmissio lies; seve geerators at buses 1, 2, 3, 6, 8, 9, 12; ad 15 OLTC trasformers The reactive power sources are cosidered at bus o 18, 25 ad 53 Lie data, bus data ad the miimum ad maximum limits o cotrol variables ad depedet variables have bee adapted from official IEEE stadards ad values The total system active ad reactive power demads are pu ad 3364 pu o 100 MVA base The voltages of all load bus ad geerator bus have bee costraied withi limits of 094 pu to 106 pu Fig 2 Sigle-lie diagram of the IEEE 57 bus test system [7] Copyright to IJAREEIE DOI: /IJAREEIE

5 Vol 4, Issue 11, November 2015 II LITERATURE SURVEY Ta et al (2013) preseted a paper applyig the Newto-Raphso method based o curret ijectio ito the case of distributio etwork Firstly, the correctio equatios of these two methods have bee derived ad compared The Jacobia matrix of the traditioal Newto-Raphso must be recalculated i each iteratio, while the Newto-Raphso method based o curret ijectio oly eed recalculate the diagoal elemets of its Jacobia matrix which maily cosists of admittace matrix s elemets It reduces the computatio ad makes the programmig easier as well Based o these, their covergece properties have bee derived Both of them have the quadratic covergece; the oly differece is the coefficiets IEEE11bus system has bee used to test this method, ad compared with the traditioal Newto-Raphso method The results show the Newto-Raphso (N-R) method based o curret ijectio reliable ad effectively The distributio etwork IEEE11 case shows that with the same iitial values, the curret ijectio methods ca coverget to the upper part of the PV curve, that is the stable solutio of the system The IEEE11 case idicates the curret ijectio method is more reliable tha the traditioal Newto-Raphso [8] Wag et al (2012) exteded research o use of Newto Raphso method for load flow studies The traditioal methods, such as electricity method, the RMS curret method ad so o, i the calculatio i the process of lie loss is very depedet o load output curve of statioary coditio But power grid operatio process load curve are chagig Therefore, use commo method to calculate the lie loss there is greater error However, use improved Newto's method for grid loss calculatio, because the data collectio process of fully cosiderig load chage The method to calculate the power loss results more close to the statistical eergy loss, so that ca effectively cotrol the size of maagemet eergy loss Therefore, this method ca be used as a ew method of distributio eergy loss calculatio [9] Bijwe et al (2003) presets ew o diverget costat Jacobia Newto power flow methods Both coupled ad decoupled Jacobia versios have bee developed The o divergece feature of these methods is achieved through applicatio of optimal multiplier theory for step size adjustmet cotrol I order to verify the effectiveess of these methods results for four IEEE test systems, two Idia power systems ad a famous 11-bus ill-coditioed distributio system have bee obtaied ad compared with those obtaied with the correspodig versios of the covetioal power flows [10] IIMETHODOLOGY I preset study, load flow study of IEEE 57 bus system has bee doe usig Newto Raphso load flow algorithm The MATPOWER software has bee used to ru the algorithm To perform load flow aalysis usig Newto Raphso load flow method, the algorithm developed is as follows: To perform load flow aalysis usig Newto Raphso method, the algorithm developed is as follows: Step 1: Form the odal admittace matrix (Yij) Step 2: Assume a iitial set of bus voltage ad set bus as the referece bus as: V i = V i, spec 0 0 (at all PV buses) V i = (at all PQ buses) Step 3: Calculate the real Power P i usig the load flow equatio; 2 P G V V V G cos B si i ii i i j ij ij ij ij j1 Step 4: Calculate the reactive Power Q i usig the load flow equatio; 2 Q B V V V G si B cos i ii i i j ij ij ij ij j1 Step 5: Form the Jacobia matrix usig sub-matrices H, N K ad L Step 6: Fid the power differeces ΔP i ad Δ Q i for all i=1, 2, 3 (-1); P i = P i,spec P i,cal Q i = Q i,spec Q i,cal Step 7: Choose the tolerace values Copyright to IJAREEIE DOI: /IJAREEIE

6 Vol 4, Issue 11, November 2015 Step 8: Stop the iteratio if all ΔP i ad ΔQi are withi the tolerace values Step 9: Update the values of V i ad δ i usig the equatio x k+1 =x k + Δx k I cotext to various steps ivolved i carryig out load flow studies with Newto Raphso method, followig detailed flow chart has bee desiged: Fig 3 Detailed flow chart of Newto Raphso load flow method IIIRESULTS AND DISCUSSION Power flow results of IEEE 57 bus system iclude voltage magitudes, active ad reactive powers ad various lie losses The results obtaied ca be used to carry out further system studies Various results obtaied usig MATPOWER are show i tables below Copyright to IJAREEIE DOI: /IJAREEIE

7 Vol 4, Issue 11, November 2015 Table 1 Load flow results for IEEE 57-bus system usig N-R method Bus Voltage Geeratio Load Mag(pu) Ag(deg) P (MW) Q (MVAr) P (MW) Q (MVAr) Copyright to IJAREEIE DOI: /IJAREEIE

8 Vol 4, Issue 11, November Total: Various load flow losses ca also be computed usig N-R method which is tabulated as below: Load flow losses for IEEE 57 bus system usig N-R method Brach From To From Bus Ijectio To Bus Ijectio Power loss Q Bus Bus P (MW) Q (MVAr) P (MW) (MVAr) P (MW) Q (MVAr) Copyright to IJAREEIE DOI: /IJAREEIE

9 Vol 4, Issue 11, November Copyright to IJAREEIE DOI: /IJAREEIE

10 Vol 4, Issue 11, November Total: V CONCLUSION I this paper, a IEEE 57 based bus system for power flow is beig discussed I this experimet Newto Raphso method is studied ad discussed to evaluate the load flow coditios icludig power losses for this bus system This techique is studied usig MATPOWER simulatio software ad the results showed faster covergece with reliable results REFERENCES [1] Zia W ad Alvarado F L, Iterval arithmetic i power flow aalysis, IEEE Trasactios o Power System, Vol 7, o 3, pp , 1992 [2] Garcia P A N, Pereira J L R, Careiro S, Vader M da C, ad Martis N, Three-Phase Power Flow Calculatios Usig the Curret Ijectio Method, IEEE Trasactios o Power Systems, vol 15, pp , 2000 [3] Su D, I, Ashley B, Brewer B, Hughes A, ad William F T, "Optimal power flow by Newto approach", IEEE Trasactio o Power Apparatus ad Systems, vol 10, pp , 1984 Copyright to IJAREEIE DOI: /IJAREEIE

11 Vol 4, Issue 11, November 2015 [4] Ambriz P, H, Advaced SVC models for Newto-Raphso load flow ad Newto optimal power flow studies, IEEE Trasactios o Power Apparatus ad Systems, vol 15, pp , 2002 [5] Stagg G ad El-Abiad A, Computer Methods i Power System Aalysis, Tata McGraw-Hill, New York, 1968 [6] Heydt G, Computer Aalysis Methods for Power Systems, Macmilla Publishig, New York, 1986 [7] Paosya A ad Oswald B, Modified Newto-Raphso load flow aalysis for itegrated AC/DC power systems," i Proc Uiversities Power Egieerig Coferece, UPEC 2004, vol 3, pp , 2004 [8] Ta T, Li Y, Li Y, ad Jiag T, The Aalysis of the Covergece of Newto-Raphso Method Based o the Curret Ijectio i Distributio Network Case, i Proc Iteratioal Joural of Computer ad Electrical Egieerig, vol 5, pp , 2013 [9] Xiao mig Wag ad Zhag Li, Improved Newto s Method for Calculatio of Power Distributio Network Theory Eergy Loss, i proc 2d Iteratioal Coferece o Computer ad Iformatio Applicatio, vol6, pp , 2012 [10] Bijwe P R, Abhijith B ad Raju G KV, Robust Three phase Newto Raphso Power Flows, i Proc Fifteeth Natioal Power Systems Coferece (NPSC), IIT Bombay, pp , 2008 Copyright to IJAREEIE DOI: /IJAREEIE