Fast Power Flow Methods 1.0 Introduction

Transcription

1 Fast ower Flow Methods. Itroductio What we have leared so far is the so-called full- ewto-raphso R power flow alorithm. The R alorithm is perhaps the most robust alorithm i the sese that it is most liely to obtai a solutio for touh problems, which are problems that start from uesses that are ot close to their solutio. For example, solvi a lare power flow case from a flat start is usually cosidered to be a touh problem, ad as a result, it is best to do that with a R. ut R is slow! Ofte, the problem is ot so touh, ad i that case, the so-called fast decoupled FDC alorithm is also effective i etti the solutio, there is o loss of accuracy, ad it is much faster. A commo situatio where FDC is attractive is whe you have solved the case, ad the you wat to re-solve the case usi a hot start to aalyze the effect of some ot-sodramatic chae. Here, the fact that the problem is ot so touh calls for relaxi solutio alorithm robustess.

2 There are situatios where speed is paramout, but accuracy is ot. For example, i o-lie aalysis of 5, cotiecies, we may wat to oly filter the cotiecies that have potetial to result i problems, ad the perform full aalysis o those. I such cases, the DC power flow is appropriate. Althouh DC power flow is fast ad robust, it is ot very accurate. Solvi the power flow equatios ca be computatioally itesive. I these otes, we review FDC ad DC power flow methods. You ca see the relatio betwee the R ad these two i Table, i terms of speed, accuracy, ad solutio robustess. Table Solutio method Speed Accuracy Solutio robustess ewto-raphso Slow Accurate Robust Fast decoupled Fast Accurate Less robust DC ery fast Approximate Robust. The fast decoupled power flow The acobia matrix,

3 has a special characteristic i that the elemets of the off-diaoal submatrices θ ad are usually very small relative to the elemets of the diaoal submatrices θ ad I fact, we have see that i the first iteratio of a flat start, whe we assume all ales are, the elemets of the off-diaoal matrices are all. This is because every term i the expressios of the off-diaoal blocs are a multiplied by a G or b multiplied by a si. The overall terms are small because, for trasmissio: coductace G teds to be small ad aular differeces across circuits ted to be small, resulti i small si terms. These observatios are cosistet with our uderstadi that is ot very sesitive to voltae maitude i.e., small x ad x ot very sesitive to ale i.e., small x., ad is x ad We ca tae advatae of these observatios i the followi way. Istead of usi the exact acobia, let s assume that all elemets of the off-diaoal submatrices are i fact ad remai throuhout the etire R alorithm. I other words, let s ust use the followi acobia:

4 T ote what this does to our update equatio: x Substituti ito, we have: T 4 erformi the idicated matrix multiplicatio, we obtai: 5 6 Equatios 5 ad 6 have the followi remarable feature: real power equatios are decoupled from the voltae maitudes, ad reactive power equatios are decoupled from the ales. The implicatio is that either oe of equatios 5 ad 6 may be solved idepedet of the other oe! 4

5 Our power flow alorithm remais exactly as it was before, with the oly exceptio bei i Step 4.. Specify: All admittace data series Y, chari capacitace, trasformer taps, & shuts d ad d for all buses whether,, or swi ad for all buses for swi bus, with =. Set the iteratio couter =. Use oe of the followi to uess the iitial solutio. Flat Start: =. for all buses. Hot Start: Use the solutio to a previously solved case for this etwor.. Compute the mismatch vector for x, deoted as fx. I what follows, we deote elemets of the mismatch vector as ad correspodi to the real ad reactive power mismatch, respectively, for the th bus which would ot be the th elemet of the mismatch vector for two reasos: oe reaso pertais to the swi bus ad the other reaso to the fact that for type buses, there are two equatios per bus ad ot oe. This computatio will also result i all ecessary calculated real ad reactive power iectios. erform the followi stoppi criterio tests: If < for all type & buses ad If < for all type buses, The o to step 5 Else Go to step Fid a improved solutio as follows: Evaluate the acobia at x. Deote this acobia as Solve for x by applyi LU decompositio to: This is oly chae i alorithm!! Compute the updated solutio vector as x + = x + x. Retur to step with =+. 5. Stop. 5

6 How to see that the FDC alorithm is faster tha R? I R, step 4 computes x The acobia has dimesio -- G. I FDC, θ has dimesio -, ad has dimesio - G. For example, if we have buses ad eerators, the the acobia i the R has dimesio of 7,999, but the FDC alorithm acobias will have dimesios of 9,999 ad 8,, respectively. It is ow that the speed of LU decompositio is a fuctio, approximately liear, of the umber of elemets. The umber of elemets i R is 7,999 =.44E9, whereas i FDC it is 9,999 +8, =7.4E8. Therefore, FDC will be about twice as fast per iteratio as R. However, because the acobia ives the directio to move the solutio i each iteratio, we do suffer a loss i accuracy per iteratio, ad therefore we may eed more iteratios to obtai the fial solutio. 6

7 Give these two opposi forces less time per iteratio ad more iteratios, it is usually the case that FDC is betwee.5 ad times faster tha R. ut what about accuracy? We have said that the FDC alorithm will be less accurate per iteratio. Does that imply that it will provide a less accurate solutio oce it stops iterati? The aswer to this questio depeds o the stoppi criterio. ote that i the above FDC alorithm, the stoppi criterio is ive i Step, ad it is exactly the same as the stoppi criterio used i the R. That is, both alorithms are computi the mismatch as x ad x, ad x ad x are computed with the full real ad reactive power flow equatios, respectively, i both alorithms. It is very importat to recoize that the approximatio i FDC alorithm is applied to the acobia matrix but OT the power flow equatios used to compute the elemets of the mismatch vector. The coclusio that we ca mae here is that A OWER FLOW SOLUTIO OTAIED Y FDC IS UST AS ACCURATE AS A OWER FLOW SOLUTIO OTAIED Y R. 7

8 . FDC alorithm: ehacemets We may simplify the FDC alorithm still further, mai it still faster but less accurate per iteratio by wori with the expressios of the acobia elemets for θ ad. Cosider the terms. If we elect G ad uder small ale approximatio so that siθ -θ ad cosθ -θ : x G ow cosider the terms x G x si. si cos cos 7 8 Aai, usi small G ad small ale approximatio, the above is x 9 We will ow mae use of a assumptio that the voltae profile is flat, i.e., =. The 9 becomes: 8

9 9 x ow cosider the summatio i the curly bracets ,, Recall that from defiitio of Y-bus elemets: : =-b b =- =: =b +b + +b,- +b +b,+ + +b ad usi the relatio from the first bullet: = ,- +b -,+ - - where b is sum of all shut susceptace at bus. Substituti this last expressio for ito, we obtai: b b ,,,, Substituti ito, we obtai: b x 4

10 We could perform the subtractio i 4 usi to see that the term is ust the eative of the sum of all o-shut braches coected to bus. However, b is typically very small compared to so that electi b is quite accurate, resulti i: x 5 Liewise, uder assumptios of flat voltae profile ad small ale, we ca show that: x x G si x cos Summarizi eqs. 7, 5, 6, ad 7, we have: x x x x

11 oti that the acobia matrix has o equatios or variables for bus the swi bus, we defie the matrix as: ' where this matrix may be obtaied from the Y-bus by simply strippi off the first row ad first colum assumi the swi bus is # ad tai the imaiary part of all elemets. This matrix is appropriate for writi the θ terms of 7 ad 5 i compact otatio, as ive by 9 below: where 9 Observe that is pre-multiplied ad post-multiplied by [] to accout for the product of two voltaes i 7 ad 5. ow, reardi the terms

12 If we could assume that we would have reactive power flow equatios for all buses i the etwor, the we would use for the terms as well. ut we do ot have reactive power flow equatios for the buses, oly for the buses. To accout for this, we eed to elimiate the rows ad colums correspodi to type buses from ad from []. Usi the umberi scheme,, G, as bei the voltae cotrol buses, ad bus +,, as bei the type buses, the elimiate row ad colum umber,, - from ad []. Let s refer to the resulti matrices as ad [ ]. The 6 ad 7 become: Summarizi, our decoupled pf equatios are: 9 ow let s loo at our correctio formula

13 Recalli eqs. 5 ad 6: 5 6 ad substituti 9, ito 5-6, we obtai: Multiplyi both sides by - results i: 4 5 There are two more chaes which prove useful i terms of capturi additioal computatioal efficiecy more speed. The first chae is a approximatio: let the secod [] i eq. 4 be the idetity matrix based o the assumptio that we have a flat voltae profile all the same ad that all voltaes are close to.. Mai this chae i eq. 4, our correctio equatios become: 6 5

14 4 The secod chae is to pre-multiply 6 by [] - ad 5 by [ ] -. This chae results i 7 8 Cosideri 7, sice [] is diaoal, [] - is 9 Multiplicatio of 9 by the real power mismatch vector ives the riht-had-side of 7: A similar thi ca be doe for the reactive power correctio equatio 8:

15 5 Multiplicatio of by the reactive power mismatch vector ives the riht-had-side of 8: ased o ad, eqs. 7 ad 8 become:

16 6 ~ ~ 4 where the otatio of the far riht-had-side i ad 4 idicates the riht-had-sides of ad. Two commets remai:

17 . Where s the speed-up? We still retai the speed up of the previous FDC alorithm, which is due to the fact that the LU-decompositio is faster per iteratio as a result of the decoupli ad correspodi reductio i total matrix elemets. The method described here provides additioal speed-up from two sources: The -matrix eed ot be reevaluated i each iteratio, ad the matrix is formed by simply deleti appropriate rows ad colums from, ad so we save the time of evaluati acobia matrix elemets. ecause the left-had-side of eq. is costat, we eed perform LU-decompositio for this equatio oly oce. Give the L ad U factors, we eed to oly perform forward ad bacward substitutio for each differet rihthad-side. We are ot quite as fortuate with the reactive power correctio equatio, 4, because there we must re-factorize each time the list of voltae cotrol buses chaes.. Alorithm: I idicated the power flow alorithm is exactly the same as i the R, but there is a mior differece i that Steps ad 4 pae 5 above ca be alterated, as follows: 7

18 a. Step a: Compute mismatch of ~ usi ad. b. Step 4a: Solve eq. for. c. Step b: Compute mismatch of ~ usi ad. d. Step 4b: Solve eq. 4 for e. Step c: erform stoppi criterio tests: If < for all type & buses ad If < for all type buses, The o to step 5 Else. Retur to step with =+. The reaso why this improves speed is because the update o voltaes are doe usi a step based o the most recet update o ales, ad this teds to reduce the ecessary umber of iteratios. 8

19 9 4. DC ower Flow Retur to equatio, repeated here for coveiece: ~ ' ow assume all voltaes are.. The eq. becomes:... 5 So, equatio 5 becomes: 6

20 This equatio, whe solved ust oce = for Δθ, ad with a flat-start solutio, implies that θ=+δθ =Δθ, ad, if we assume that this solutio is the correct oe, the i other words, 7 where are the real power flow equatios for buses to evaluated at θ = i.e., the flat start, ad are the real power flow iectios ito each bus to. Therefore, 8 This ives all of the ales for the etwor with a sile solutio to a set of liear equatios. The, the real power flows ca be computed with 9 which is the power flowi across a circuit coected betwee buses ad uder coditios of a electi resistace, b small ale approximatio, ad c all voltae maitudes are..