Term Project - select journal paper and outline. Completed analysis due end

Transcription

1 EE Lecture 30 Fr ov 4, 2016 Topcs for Today: Announcements Term Project - select journal paper and outlne. Completed analyss due end of Week 12. Submt va e-mal as mn-lecture.ppt wth voce narraton. Software: onlne students - apply for ATP/ATPDraw lcense, verfy lcensng when you receve t by e-mal, and we wll mal you the nstall CD. ASPE software - runs off of MTU server va nternet. Offce: EERC 614. Phone: W,F - EERC 101, 3-4pm, Sat.. Recommended problems & all solutons: Ch.9 solns now posted. Upcomng materal: balanced 3-phase ssues: powerflow, operaton. After that: Unbalanced 3-phase short crcut analyss. Chapter 9 - Powerflow [Ybus] used for powerflow does not nclude orton generator admttance. Generator vewed as an deal source of P and Q. P-Q formulaton usng [Ybus] - See RFL.pdf handout Runnng a powerflow - slack, P-Q (or load), P-V (or gen) buses Smulaton output - lne flows, generator output, lmts, etc. Lne loadng lmts

2

3

4

5

6

7

8

9

10

11

12 EWTO-RAPHSO LOAD FLOW FORMULATIO Dr. Bruce Mork EE Fall 2016 At a gven bus n the system, there can exst: Fxed P and Q njecton consstng of: < Scheduled generaton that njects P G nto the bus. < A fxed load of P L + jq L (an njecton of - P L - jq L ) P and Q flowng nto bus from the network (all part of [Y BUS ]): < Transmsson lnes - short, medum, long; sngle-crcut, double-crcut where mutual couplng s neglected, or double-crcut wth mutual couplng effects. < Transformers - 2-wndng or 3-wndng; fxed rato, LTC, or Phase-Shftng. < Shunt reactors: Y = 1/(jωL) = - jb REACT < Shunt capactor banks: Y = jωc = jb CAP < A voltage-dependent load represented as a shunt admttance: Y LOAD = G + jb. Bus 1 Sched Gen P G+ jq G P L+ jq L Sched Load Bus + ~ V - ~ I Bus 2 P T+ jq T Bus ETWORK AS DEFIED BY Y BUS LIES, XFMRS, SHUT CAPS, REACTORS REF Important thngs to note: < The scheduled generaton P G s dctated by the system dspatch center va SCADA. The generator s governor s gven a set pont and holds P G constant wthn a close tolerance. Also, the generator s excter holds the bus voltage V at a constant magntude (ts angle δ s not drectly controlled and s an unknown). < The fxed load P L + jq L represents the aggregate load suppled to local consumers. In plannng studes, ths s usually a worst-case projecton of what planners thnk the load wll be 5 or 10 or more years nto the future. < P T and Q T are the total P and Q flowng ITO the transmsson grd defned by [Y BUS ]. Ths ncludes the effects of shunt capactor banks and reactors.

13 When formng equatons, t s extremely mportant to establsh a reference drecton for the flow of P, Q, and current. Ths s clearly labeled on the sketch on the ~ precedng page. Recall that the current s the net current njected nto the I network at bus by the generator and load (ths s the same njected current that occurs n the equaton [Y BUS ][V] = [I] ). Bus voltages are measured wth respect to the same reference that [Y BUS ] s referred to. otatons: The voltages and currents we are dealng wth are RMS phasor values. In the equatons we develop, t s necessary to refer to ther magntudes and angles. For example, the voltage at bus k wth respect to reference s: RMS phasor value: ~ V k or V k or V k /δ k ~ ~ RMS magntude: V k or just V k Angle of Vk : δ k We also need to refer to ndvdual elements of [Y BUS ]. The entry n the,j poston s a complex number wth a magntude of and an angle of y j y j j The Setup: At each bus, there are just three components to the P and Q beng njected. If we follow the development of Heydt s book, we wll consder the summaton of P and Q nto a gven bus (refer to the fgure on the prevous page and be sure to get the sgns rght). When the system s n equlbrum the total P and total Q flowng nto the bus wll be zero. PITO BUS PG PL PT 0 QITO BUS QG QL QT 0 Observe that P T and Q T are functons of the bus voltages, whle P L and Q L and P G are constants. Q G s a slack varable (more on t later). ote that these two equatons together make up the nonlnear functon F(δ,V) = 0 whch wll be solved wth ewton-raphson teraton. (.e. ntal guesses for the unknown V s and δ s are made and an teraton s performed that drves the V s and δ s toward values that make F(δ,V) = 0). When the teraton has converged, we know all of the bus voltages n the system and thus can calculate all P and Q flows through transmsson lnes and transformers.

14 In order to calculate P T and Q T we must frst know the value of ~ ~ I y V j j j 1 ~~ * ~ ~ ST PT jqt V I V y jvj j 1 PT V y jvj cos( j j) j 1 QT V y jvj sn( j j) j 1 * ~ I, whch can be found by multplyng row of [Y BUS ] tmes the bus voltage vector. In the form of a summaton, t s: The complex power flowng nto the network at ths pont s thus Resolvng t nto ts real and magnary components, Thus, the total P and Q flowng nto bus for a converged soluton s F (, V) P P P P V y V cos( ) P, I, ITOBUS G L j j j j j 1 F (, V) Q Q Q Q V y V sn( ) Q, I, ITOBUS G L j j j j j Heydt lumps load and generaton together: P = P - P and Q = Q - Q G L G L and refers to them as specfed actve and reactve powers. The msmatches ΔP and ΔQ are defned as the dfference between the specfed P and Q (flowng nto the bus from the load and generator) and the P and Q flowng out of the bus and nto the network. At equlbrum (when loadflow has converged) the msmatches are, wthn a tolerance of ε, equal to zero. However, durng the teraton, the msmatches are nonzero and are a functon of the present values of δ and V. At teraton step m, m m m m m P P P V P P V P P V T(, ) T(, ) T m Q m m m m Q Q V Q Q V Q Q V T (, ) T T (, ) (, ) (, ) The complete expressons for the msmatches at teraton step m are thus gven as: m m m m m P P V yjvj cos( j j) j 1 m m m m m Q Q V yjvj sn( j j) j 1

15 ote that Heydt has a sgn error n the way that he defnes ΔP and ΔQ (see p.149), but then recovers from t by taggng a mnus sgn on [J] (see p.150). The complete formulaton for the loadflow s thus n the form J P V Q ΔP s the column vector of P msmatches at all buses except the slack bus. ΔQ s the column vector of Q msmatches at all load buses (Q s a slack varable at all generator buses and at the slack bus and so these buses are not ncluded). [J] s the Jacoban matrx contanng the partal dervatves of the expressons for P and Q flowng nto each bus. These partal dervatves fall nto 4 categores and [J] s often parttoned nto 4 submatrces descrbed as follows: or J1 J2 P J3 J4 V Q PI QI P V Q V I I P V Q The partals can be obtaned from the equatons for P I and Q I. They are lsted n equatons (4.38) through (4.45) n your text. For the man dagonal terms of J1 note that when j =, δ - δ j = 0 and the partal s 0. J1 P I, V y jvjsn( j j) j 1 For the off-dagonal terms of J1, only one of the terms of the summaton has a nonzero partal dervatve: PI, J1k VV k yk sn( k k) k k For the man dagonal terms of J2, j = so P I, J2 yjvjcos( j j) 2Vy cos( ) V j 1 VV V 2 j whch leads to

16 For off-dagonal terms of J2, J3 PI, J2k V yk cos( k k) k V k For man dagonal terms of J3 (note sgn error n equaton 4.42): For off-dagonal terms of J3: For man dagonal terms of J4: Q I, V y jvjcos( j j) j 1 QI, J3k VV k yk cos( k k) k k Q I, J4 yjvjsn( j j) 2Vy sn( ) V j 1 Fnally, for off-dagonal terms of J4: QI, J4k V yk sn( k k) k V k All terms n the Jacoban and n the msmatch vector are evaluated usng present values of V and δ. The column vector for Δδ ΔV s then solved. Typcally ths s done usng sparse matrx data structures and some type of n stu LU factorzaton. The values of δ and V used n the present teraton are then updated: m1 m m V V V Convergence s usually determned by montorng the msmatch vector. The norm of the msmatch vector could be used as a convergence measure but usually s not. Testng for ΔP # ε at all PQ and PV busses, and testng for ΔQ # ε at all PQ buses s done. Choosng ε = per unt s common. Typcally, the Q msmatches are greater than P msmatches so convergence often depends on ΔQ. If the precson of the loadflow study s not of prmary concern, the convergence tolerance for Q s sometmes relaxed to 10 ε or else the condton s modfed to be ΔQ 2 # ε.