Chapter - 2. Distribution System Power Flow Analysis

Transcription

1 Chapter - 2 Dstrbuton System Power Flow Analyss

2 CHAPTER - 2 Radal Dstrbuton System Load Flow 2.1 Introducton Load flow s an mportant tool [66] for analyzng electrcal power system network performance. Load flow s used at the tme of solvng optmzaton problem. So, load flow should be fast and effcent. The dstrbuton network s radal n nature havng hgh R/X value whch makes t ll condtoned. The conventonal Gauss Sedel (GS) and Newton Raphson (NR) method does not converge for the dstrbuton networks. As dscussed n chapter 1(secton ), several lmtatons exst n radal load flow technques presently reported n the lterature such as complcated bus numberng schemes, convergence related problems, and the nablty to handle modfcatons to exstng dstrbuton system n a straght forward manner. Ths motvated the development of a radal dstrbuton load flow soluton method. Most of the conventonal load flow methods consder power demands as specfed constant values. Ths should not be assumed because n dstrbuton system bus voltages are not controlled. Loads are specfed by constant power, current or mpedance requrements. There are several load flow methods based on backward forward sweep technque. These method bascally use - 1. Current summaton methods 2. Power summaton methods 18

3 Current summaton method s more convenent and faster than the Power summaton method because t uses only V and I nstead of P and Q. Bascally, load flow s used for the determnaton of the dstrbuton system operatng pont at steady state under gven condtons of loads. Frst of all, all bus voltages are calculated. From these bus voltages, t s possble to drectly calculate currents, actve & reactve power flows, actve & reactve system losses and other steady state quanttes. In ths chapter, a backward forward sweep method s presented for solvng the load flow problem of a dstrbuton system. The mathematcal formulaton of the above method s explaned n Secton 2.2. In ths Secton, dervaton of voltage, angle, actve and reactve power losses, voltage devaton and voltage stablty ndex from sngle lne dagram of dstrbuton system are dscussed. The steps of load flow algorthm calculaton are presented n Secton 2.3. In Secton 2.4, the effectveness of the proposed method s tested wth 33 and 69 bus radal dstrbuton systems and the results are compared wth the exstng methods. In Secton 2.5, conclusons of the proposed method are presented. 2.2 Mathematcal Formulaton Assumpton In ths secton, crcut model of radal dstrbuton system (RDS) s presented. It s assumed that the three phase radal dstrbuton system s balanced and can be represented by an equvalent sngle lne dagram. The lne shunt capactance at dstrbuton voltage level s neglgbly small and, hence neglected. The voltage magntude and phase angle of the source should to be specfed. Also, the complex values of load demands at each bus along the feeder should be gven. Intally, a flat voltage profle s assumed at all buses.e., 1.0 pu. The Fg. 2.1 shows the sngle lne dagram of n-bus RDS. Fg. 2.1: Sngle - lne dagram of n-bus RDS 19

4 2.2.2 Bus ndexng (numberng) scheme Generally, most load flow formulatons are the set of equatons and unknowns, assocated wth ndvdual buses. So the formulatons are arranged by approprate bus ndexng. But n radal dstrbuton system, the equatons and unknowns can be reduced such that the equatons and unknowns become correspondng to the laterals. So lateral ndexng s needed for radal dstrbuton system load flow. A radal dstrbuton system can be structured as a man feeder wth laterals and these laterals may also have sub- laterals, whch themselves may have sub- laterals, etc. Each bus s assgned an ndex (l, m, n ), where l, m and n corresponds to level of lateral, lateral ndex and bus ndex respectvely. The reverse breadth-frst (RBF) orderng of the laterals, found by sortng the lateral ndces n descendng order, frst by level, then by lateral ndex. The RBF orderng s typcally used for backward sweep type operatons. If the laterals are sorted n ascendng order, the result s a breadth-frst (BF) orderng, typcally used for forward sweep type operatons Backward and Forward sweep based load flow algorthm Iteratons of backward and forward sweep based load flow algorthm: 1. Frst the bus and lateral ndexng are done. 2. Then RBF orderng of end nodes are done accordng to the ndexng. 3. All the end nodes voltages are ntalzed for the three phases (consderng the nomnal voltage as base voltage). 4. Accordng to the RBF orderng the backward sweep for the frst teraton starts. 5. In backward sweep (a) (b) (c) The end node of lowest RBF order s consdered. As the node voltage s known, so the current njectng at ths node by loads, shunt capactors and DGs can be calculated. Then the current njecton at the node s calculated by applyng KCL at current nod. As ths s the end node, no current should be added by the ncomng downstream branch. 20

5 (d) (e) (f) (g) (h) Then by the ntalzed voltage and the total current njecton at the current node, usng the update formulae, voltage at the next node and the current njecton at the next node through the branch between current node and next node are calculated. Go to 5(b) and follow the same process untl the branch off node of current sub lateral or lateral has reached. The calculaton of voltage and current at the current sub lateral or lateral are stopped and the RBF order s ncremented by one and goes to 5 (b) untl source node has reached. If the node s source node, and RBF order value becomes maxmum, t means that the backward sweep for frst teraton ends. So from backward sweep the branch off current of laterals and sub laterals are stored for computng the node voltages n forward sweep. 6. In forward sweep - (a) (b) (c) (d) The forward sweep starts wth the specfed substaton secondary voltage, and the current njected by the substaton to the network whch was stored as branch off current of the man feeder durng the backward sweep. It s therefore easy to calculate the voltage of the downstream node and the current flowng through the downstream branches of the man feeder one by one usng the update formulae (and the branch off current stored durng backward sweep and the branch off voltages stored durng the calculaton of the prevous man feeders or laterals s used except for man feeder calculaton) untl the end node of that man feeder or laterals or sub laterals reaches. Durng ths, voltages at the ponts from where the laterals are branchng off from ths man feeder or laterals or sub laterals are stored as branch off voltages of correspondng laterals. Now f the end node of that man feeder or the lateral or the sub lateral reaches, RBF order s decreased by one and go to 6 (b) untl the calculaton of the lateral of RBF order one reaches. 21

6 7. After completon of the backward and forward sweep for the frst teraton, end node voltages are updated. 8. Ths new end node voltage s compared wth the prevous ntalzed end node voltages (only for frst teraton) or the end node voltages of the prevous teraton. 9. If these compared end node voltages values are less than a small error value, t means that the load flow has converged otherwse go to 5 and do the backward and forward sweep repeatedly consderng the new end node voltages Load Flow Calculaton The load flow of a sngle source network can be solved teratvely from two sets of recursve equatons. The frst set of equatons for calculaton of the power flow through the branches by startng from the endng buses and movng n the backward drecton towards the source bus (substaton bus). The other set of equatons are for calculatng the voltage magntude and angle of each node startng from the slack bus (substaton bus) and movng n the forward drecton towards the endng bus. These recursve equatons are derved as follows. The fg. 2.1 shows the sngle lne dagram of n-bus radal dstrbuton system. Consder a branch j s connected between the buses and j Backward sweep The updated effectve power flows n each branch are obtaned n the backward sweep computaton by consderng the bus voltages of prevous teraton. It means the voltage values obtaned n the forward path are held constant durng the backward sweep and updated power flows n each branch are transmtted backward along the feeder usng backward path. Ths ndcates that the backward sweep starts at the extreme end bus and proceeds towards source bus (substaton bus). The actve and reactve power flows are calculated n backward drecton. The effectve actve ( P ) and reactve ( Q ) powers that of flowng through branch j from node to node j can be calculated backwards from the endng bus and s gven as, P P j P L j j ' 2 ' 2 ( P j + Q j ) = + + R (2.1) 2 V 22 j ' 2 ' 2 ( P j + Q j ) Q = Q j + Q L j + X j (2.2) 2 V j

7 Where ' P j = P j + P Lj and ' j Q = Q + j Q Lj P Lj and Q Lj are loads that are connected at bus j P j and Q j are the effectve real and reactve power flows from bus j Forward sweep The purpose of the forward sweep s to calculate the voltages at each bus startng from the feeder source bus (substaton bus). The feeder substaton voltage s set at ts actual value. Durng the forward propagaton the effectve power n each branch s held constant to the value obtaned n backward walk. The voltage magntude and angle at each bus are calculated n forward drecton. Consder a voltage V δ at bus and V j δ j at bus j, then the current flowng through the branch j havng an mpedance, Z j = R + j X connected between j j and j s gven as, I j = ( V δ V j δ j) ( P jq) And I j = V R j + δ j X On equatng the equaton (2.3) and (2.4), we have ( P jq ) ( V δ V j) j δ δ = V R j X j j + j ( j ) ( jq )( jx ) V = P R + 2 V V j δ δ j j (2.3) (2.4) (2.5) (2.6) By equatng real and magnary parts on both sdes of equaton (2.6), we have 2 ( δ δ ) ( P ) V V cos = V R + Q X (2.7) j j j j V V j sn( δ j δ ) = Q R j P X (2.8) j Squarng and addng equatons (2.7) and (2.8), we get 2 ( j) ( Q ) = V R j+ X j + Q R j X j V V P P ( V ) ( ) ( 2 2 )( 2 2 V j V V P R j Q X j R j X j P Q) (2.9) = (2.10) 23

8 ( ) ( ) ( ) P + Q R Q j X j = V V V 2 P R X j j j 2 (2.11) and voltage angle, δ j can be derved on dvng equatons (2.8) and (2.7) tan δ j ( δ j δ ) = = δ + tan Q R P X j j ( P Q ) V R X 1 2 j + j Q R P X j j ( P Q ) V R X 2 j+ j (2.12) (2.13) The magntude and the phase angle equatons can be used recursvely n a forward drecton to fnd the voltage and angle respectvely of all buses of radal dstrbuton system Convergence crteron The voltages calculated n the prevous and present teratons are compared. In the successve teratons f the maxmum msmatch between the voltages s less than the specfed tolerance.e., , the soluton s sad to be converged. Otherwse new effectve power flows n each branch are calculated through backward walk wth the present computed voltages and then the procedure s repeated untl the soluton s converged Actve and Reactve power losses calculaton The actve and reactve power losses of branch j can be calculated as, P Q L o s s L o s s (, j ) (, j ) 2 2 P + Q = R j 2 (2.14) V 2 2 P + Q = X j 2 (2.15) V The total actve and reactve power loss of radal dstrbuton system can be calculated as, n 1 ( P 2 2 ) nb ( Q P + Q) (2.16) P = R = R T, Loss j 2 j 2 j= 1 V j= 1 V n 1 ( P 2 2 ) nb ( Q P + Q) (2.17) Q = X = X T, Loss j 2 j 2 j= 1 V j= 1 V 24

9 Voltage devaton calculaton The voltage devaton of the system s defned as V D n = V V (2.18) = 1, nom where V, nom V, rated 1 p. u = = (2.19) Voltage stablty ndex calculaton From Fg.2.1, current that of flowng through branch j from bus to bus j can be calculated as I j = R V j + V j j X j (2.20) ' * ' P j j Q = V j j I (2.21) j Eq. (2.22) gves the voltage stablty ndex at all buses n RDS, was proposed by Chakravorty et al [67]. Usng eq. (2.20) and (2.21) : j= V 4 j X j Qj R 4 j j Rj+ Qj j V SI P P X (2.22) SI > 0, ndcates stable operaton of RDS. j Objectve functon for mprovng voltage stablty ndex s gven by (2.23): VSI 1 = SI j, j = 2, 3,... n (2.23) The maxmum value of SI j gves mnmum value of objectve functon (VSI ). So, mnmum values of objectve functon ndcate mprovement of voltage stablty ndex.. 25

10 2.3 Algorthm for Load Flow Calculaton The flow chart for load flow s shown n fg.2.2. The backward forward sweep load flow algorthm s gven below Start Read lne & load data Set flat voltages [1 p.u] for all buses (nodes) Computer effectve real & reactve power flows of all branches usng backward sweep from equatons [2.1] & [2.2] Update bus voltages & phase angles usng forward sweep from equatons [2.11] &[2.13] No Is load flow converged Yes Compute branch power losses, total system losses, total system voltage devaton, total system voltage stablty ndex & prnt the results. Stop Fg.2.2: Flow chart for Radal Dstrbuton Load Flow 26

11 Step 1 : Read dstrbuton system lne and load data. Assume ntal bus voltages are 1 p.u and set ε = Step 2 : Start teraton count, IT =1. Step 3 : Intalze actve power loss and reactve power loss vectors to zero. Step 4 : Calculate the effectve actve and reactve power flow n each branch usng equatons (2.1) and (2.2). Step 5 : Calculate bus voltages, actve and reactve power loss of each branch usng equatons (2.11), (2.14), and (2.15) respectvely. Step 6 : Check for convergence.e., V < ε n successve teratons. max If t s converged go to next step otherwse ncrement teraton number and go to step 4. Step 7 : Calculate the actve and reactve power losses for all branches, total real and reactve power loss, voltage devaton of each bus, total voltage devaton, voltage stablty ndex of each bus and voltage stablty ndex of system. Step 8 : Prnt voltage at each node, the actve and reactve power losses of all branches, total actve and reactve loss, total voltage devaton and voltage stablty ndex of system. Step 9 : Stop. 2.4 Smulaton Results and Analyss The proposed method computes the load flow soluton for the gven RDS. The effectveness of the proposed method s tested on 33 and 69 bus RDS Test system -1 (33 bus RDS) The test system -1 s a kv, 33 bus RDS consstng of 33 buses confgured wth one substaton, one man feeder, 3 laterals and 32 branches. The total actve and reactve loads on ths system are 3715 kw and 2300 kvar, respectvely. It s demonstrated n Fg. A.1 [68]. The lne and load data of ths system s gven n appendx Table A.1 & A.2. 27

12 Table 2.1 gves results of voltage magntude and angles at dfferent buses of ths system. The voltage magntude and angles at dfferent buses of ths system are gven n Fg.2.3 & Fg.2.4 respectvely. The mnmum voltage s p.u at bus 18 and maxmum voltage regulaton s 9.58%. Table 2.1: Voltage magntudes and angles of 33 bus RDS Bus No. Voltage Magntude (p.u) Voltage Angle

13 Voltage magntude (p.u.) Bus No. Fg.2.3: Bus voltage magntude (p. u.) of each bus n 33 bus RDS Voltage angle (deg.) Bus No. Fg.2.4: Bus voltage angle (deg.) of each bus n 33 bus RDS 29

14 Table 2.2 gves results of actve power and reactve power losses of each branch of the system. The real power and reactve power losses of each branch of the system are gven n Fg. 2.5 & Fg. 2.6 respectvely. The total actve and reactve power losses of the system are kw and kvar respectvely. The actve and reactve power losses are 5.65% and 6.19% of ther total loads. Table 2.2: Power loss of 33-bus RDS Br. No. SE Bus RE Actve power loss P Loss (kw) Reactve power loss Q Loss (kvar)

15 Actve power loss ( kw ) Reactve power loss ( kw ) Total loss Branch No Fg.2.5: Actve power loss (kw) of each branch n 33 bus RDS Branch No. Fg.2.6: Reactve power loss ( kvar) of each branch n 33 bus RDS 31

16 Comparson of load flow results between the proposed method and the exstng method [69] s gven n Table 2.3. The total actve and reactve power losses are reduced and the mnmum voltage s mproved n the proposed method. Table 2.3: Comparson of load flow results of 33 bus RDS Descrpton Total actve power loss, P Loss (kw) Total reactve power loss, Q Loss ( kvar ) Mnmum Voltage and t s bus number CPU tme(s) Exstng method [69] at bus Proposed method at bus Table 2.4 gves results of voltage devaton and voltage stablty ndex of the system. The voltage devatons of each bus of the system s gven n Fg The voltage stablty ndex of each bus of the RDS s gven n Fg Table 2.4 : Voltage devaton & Voltage stablty ndex of 33 bus RDS System VD n p.u. VSI 33 bus V o lta g e de v a to n ( p.u. ) Bus No. Fg. 2.7: Bus voltage devaton ( VD ) n p.u of each bus n 33 bus RDS 32

17 1.6 Voltage stablty ndex (p.u) Fg.2.8: Voltage stablty ndex (VSI ) of each bus n 33 bus RDS Test system -2 (69 bus RDS) The test system -2 s a kv, 69 bus large scale RDS consstng of 69 buses confgured wth one substaton, one man feeder, 7 laterals and 68 branches. The total actve and reactve loads on ths system are kw and kvar, respectvely. It s demonstrated n Fg. A.2 [70]. The lne and load data of ths system s gven n appendx Table A.3 & A.4. Table 2.5 gves results of voltage magntude and angles at dfferent buses of ths system. The voltage magntude and angles at dfferent buses of ths system are gven n Fg.2.9 & Fg.2.10 respectvely. The mnmum voltage s pu at bus 65 and maxmum voltage regulaton s 8.99% Bus No. Table 2.5: Voltage magntudes and angles of 69 bus RDS Bus No. Voltage Magntude Voltage Angle

18

19 Voltage magntude (p.u.) Bus No. Fg.2.9: Bus voltage magntude (p. u.) of each bus n 69 bus RDS 35

20 0.06 Voltage angle (deg.) Bus No. Fg.2.10: Bus voltage angle (deg.) of each bus n 69 bus RDS Table 2.6 gves results of actve power and reactve power losses of each branch of the system. The actve power and reactve power losses of each branch of the system are gven n Fg & Fg respectvely. The total actve and reactve power losses of the system are kw and kvar respectvely. The actve and reactve power losses are 5.91% and 3.78% of ther total loads. Table 2.6: Power loss of 69 bus RDS Br. Bus P No. Loss (kw) Q Loss (kvar) SE RE

21

22 Real power loss (kw ) Total loss Branch No. Fg.2.11: Actve power loss (kw) of each branch n 69 bus RDS 38

23 20 Reactve power loss ( kw ) Branch No. Fg : Reactve power loss ( kvar) of each branch n 69 bus RDS The load flow results of the proposed method are compared wth the exstng method [69] n Table 2.7. The total actve and reactve power losses are reduced and the mnmum voltage s mproved by the proposed method. Descrpton Table 2.7: Comparson of load flow results of 69 bus RDS Total actve power loss, P Loss (kw) Total reactve power loss, Q Loss (kvar) Mnmum Voltage and t s bus number CPU tme (s) Exstng method [69] at bus Proposed method at bus Table 2.8 gves results of voltage devaton and voltage stablty ndex of the system. The voltage devatons of each bus of the system s gven n Fg The voltage stablty ndex of each bus of the RDS s gven n Fg Table 2.8: Voltage devaton & Voltage stablty ndex of 69 bus RDS System VD n p.u. VSI 69 bus

24 0.1 Voltage devaton (p.u.) Bus No. Fg.2.13: Bus voltage devaton ( VD ) of each bus n 69 bus RDS 1.5 Voltage stablty ndex (p.u) Bus No. 2.5 Conclusons Fg.2.14: Voltage stablty ndex (VSI ) of each bus n 69 bus RDS In ths chapter radal dstrbuton load flow method s descrbed. It s tested over two balanced radal dstrbuton systems. The proposed backward and forward sweep technque gves advantages over the other load flow technques. It does not employ complcated calculatons,.e. the dervatves of the power flow equatons. It s flexble and easly accommodates changes that may occur n any RDS. These changes could be modfcatons or 40

25 addtons of transformers, capactors, dstrbuted generators, other systems or both to the dstrbuton system. The proposed backward and forward sweep technque s easy to program and has the fastest CPU computaton tme when compared to other radal and conventonal power flow methods. Such advantages make the backward and forward sweep technque a sutable choce for plannng and real-tme computatons. The teratve technques commonly used n transmsson networks are not sutable for dstrbuton power flow analyss because of poor convergence characterstcs. It s found that the propose load flow method s sutable for fast convergence characterstcs. 41