Hands on Teaching

8/13/2019 Hands on Teaching

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Hand s -o n Teachingof Power System Dynamics

nowledge of power system dynamics is essential forK ny engineer who is to des ign or operate a moderninterconnected power system. System instability and , inthe extreme, sys tem col lapse a re very cos t ly to powerutili t ies and their cu stomers.The U.S. /Canadian interconnected power sys tem sthe largest in the world. It relies on sophisticated con-t rols and hands-on control by experienced operato rs tor e a c h t h e s t a n d a r d of s e c u r i ty t h a t i t s c u s t o m e r sdemand. I t is cont inua l ly changing and expanding tomeet consumer demand. The current t rend for indepen-dent power producers and nonutili ty generation makespower system design more difficult and a good knowl-edge of the ph enom ena of po wer sys tem s tabi l i ty andinstability more necessary.P o w e r s y s t e m d y n a m i c s is n o t a n e a s y s u b j e c t t oteach. W hat is required is a hands-on compu ter cou rse inwhich stud ents can work on simplified but, at th e sam etime, realistic models of th e powe r system . This articlefeatures a power system analysis and design package foruse in hands-on teaching of power system dynamics.I Cherry Tree Scient i fic Software

Rensselaer Polytechnic Inst i tute

stability an d modeling.Power sys tem dynamics concepts andth e neces sa ry ma them at i c s a re d i ff icu lt fo r many tog r a s p . L ar g e a n d c o m p l e x m a t h e m a t ic a l m o d e l s a r erequired for practical stability studies, and it is difficult,in th e c lass room, to get the r ight mix of practicality ands i m p li c it y . T o d e v e l o p d y n a m i c m o d e l s r e q u i r e s aknowledge of t he d evices being m odeled, their abstrac-tion in to mathematical eq uation s, and th e physical l imitso v e r w h i c h t h e e q u a t i o n s a r e v a li d . A p a r t f r o m t h eprime dynamic component of an interconnected powersys tem, the synchronous g enerator , modern power sys-tems contain many important electronic devices, eitherbeing used as control lers for the generators or on theirown, such asHVDC links an d stat ic var c om pen sators .Most power sys tem c ourses prep are the power sys-tem s tude nt wi th th e necessary background, but tend toconsider t he devices in isolation. Prospective power sys-t e m d e s i g n e rs o r o p e r a t o r s n e e d t o k n o w h o w t h e s edevices in t e rac t dynamical ly and need t o s e e wha t sinvolved in th e large simulations that are used in practi-cal utili ty environments. They n eed t o write cod e basedon t h e f u n d a m e n ta l m o d e l s of t y p ic a l p o w e r s y s t e mdynamic devices , and use that cod e to produce s imula-tion results. By running the sensitivity studies based ontheir developed code, they learn t o interpret th e s imula-

2 IEEE Computer Applications in Powe r ISSN 0895 0156/95/54.0001995 IEEE


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t i o n r e s u l t s a n d r e c o g n i z eposs ible sys tem difficult iesi n a w a y i m p o s s i b l e w i t hclassroom instruction only.Writing CodeWriting cod e for power sys-t e m s i m u l a ti o n s n o e a s yt a sk . T h e e q u a t i o n s a r ecomplicated and, even withtem m ode l s , t he r e i s a grea tdea l of d a t a t o be i nput and

Combining ?ATLAB capabilitiess impl i f i ed educa t iona l SYS-

output . Us ing com pute r l an-

with a toolbox of power systemroutines yields the right mixof practicality and simplicityg u a g e s s u c h a s F OR T R ANa n d C, i t takes cons iderable in the classroomt ime to o rganize even a s im-ple program. The mos t t roublesome aspec t s a r e no ta l w a y s in s t r u c t i v e t o t h e m a i n t h e m e , i n t h i s caseunders t anding power sys t em dynamics . A lot of t ime

s s p e n t p r o d u c i n g c o d e t o i n p u t a n d o u t p u t d a t a .Debugging the coded algori thm s also t ime consum-ing. While a useful exercise in discipl ine and c are , thet ime for unders t anding i s reduced , and cod e s oftenproduced by a team effort ra ther than by an individ-ual.MATLAB is an a id used by many unive rs i ty d i s c i -p l ines to he lp s tudent s to dev e lop a lgori thms . As i t sname sugges ts , i t is matrix based, and essentially con-s is ts of a num ber of ro utines tha t perform ma trix calcu-l a ti o n s. A s s o c i a t e d w i t h t h e s e r o u t i n e s i s a ninterpreted language with w hich the funct ions m ay becal led to perform complicated matr ix based analys is .Input and output analysis functions are available, and

bus vottage magnitude profile

c.-Q

0 20 40 60 80intemal bus numberI Figure 1. Load flow voltage prof ile at 1.315 power

increase ratio

there is a cons ide rab le bod yof a l r e a d y w r i t t e n a l g o -r i t h m s t h a t p e r f o r m m o r es p e c i a l i z e d f u n c t i o n s . I nbui lding an algori thm, t imeis c o n s u m e d in c o r r e c t i n gt h e a l g o r i t h m r a t h e r t h a nproviding pret ty, ins t ructorp leasing , i nput and output .Although it is q u i t e e a s y t ol e a r n , t h e c a p a b i l i t ie s ofMATLAB make i t no meretoy . I t s numer ica l rou t inesa r e a c c u r a t e a n d f a s t. Ina d d i t i o n , it i s n e a r l y p l a t -f o rm i n d e p e n d e n t , a n d t h esam e programs can be usedon both PCs and works tat ions .T o m a k e t h e m o s t of MATLAB in a power systemdynamics course , t ime mus t normal ly be spent by the

ins t ructor bui lding up a library of MATLAB functionsspecif ic to power sys tem dynam ics . This is neces sa rybecau se, although MATLAB simplifies coding an d al gor i thmic deve lopment , t o fu lly deve lop co de for eachp o w e r s y s t e m d y n a m i c d e v i c e a n d i t s c o n n e c t i o nthrough t he t ransmis s ion ne twork s t il l r equi res m orecomputational effort than students can afford to make.Of course , this problem of t ime s not confined to powersys tem dynam ics. In oth er fields, a num ber of collectionsof a lrea dy prog ram me d MATLAB function s (toolboxes)are avai lable . For example, in the dynamics area , thecontrol, robust control, and signal processing toolboxesfor MATLAB have proved extremely useful to both edu-cators a nd professional eng ineers.

rk Edlt YWlows JMpcritic al eigenvector0 5I I

intemal bus numberFigure 2. Eigenvector corresponding to criticaleigenvalue at 1.315power increase ratio

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Power System ToolboxT h e P o w e r S y s t e m T o o lb o x(ava il ab le f rom t he au thor )has been deve loped to g ivep o w e r s y s t e m e n g i n e e r s a na l r e a d y p r o g r a m m e d s e t ofrout ines for use wi th MAT-LAB. The func t ions can beg r o u p e d i n t o t w o g e n e r a lareas , load f low and dynam-ics.In t h e l o a d f l ow g r o u p ,there are funct ions that :Form t he power sys temn e t w o r k a d m i t t a n c ematrixo r m t h e l o a d f l o wJacobian matrix

Universities must ensurethat their power system

engineering graduates arefamiliar with the basic

principles o fpower systemstability and modeling

Calculate the mismatch eserform a New ton-Raphson load flow.In th e dynam ics group, ther e are functions that model:Generators and their controls: exciters, pow er sys-tem stabilizers, govern orsStatic var c ompe nsators.Another function is provided t hat u ses Euler s integra-t ion method to perform s tep-by-s tep t rans ient s imula-tion. RungeK utta integration is provided by a s t andardMATLAB function.Th e models can b e used for transient stability simula-tion o r for small signal stability studies. Demonstrationfiles that use t he basic functions for load flow, transien tstability analysis, and small signal stability analysis a reprovided. Th e following are tw o examples of t h e u s e oft h e Pow er Sys tem To olbox an d MATLAB for va r iousforms of power system dynamic analysis.

System Voltage StabilityUsing Modal AnalysisVol tage s t ab i li t y has been th e top ic of m any rece nt

1 o 0.1226 0.33521.25 0.1702 0.39001.30 0.2079 0.40621.31 0.2215 0 40991.32 0 2398 0.41391.34 0.3096 0.42331.35 0.4011 0.42951.355 0.5134 0 W1.36 0 9992 0.43971.361 1.6633 0.44211.3615 7.6775 0 4446

p a p e r s a n d s of grea t cur -r e n t i n t e r e s t t o p r a ct ic i n gp o w e r s y s t e m e n g i n e e r s .Despite this, our und erstand-i n g of t h e p h e n o m e n ainvolved is still not com plete.Generally, voltage instabilityin an in t e rconnec ted powersys tem can be pred ic ted byt e s t in g t h e c o n v e r g e n c e ofthe pow er flow as t he syst eml o a d a n d g e n e r a t i o n isincreased in som e way. Moreinformation on t he nature o fa n y i n st a b il it y c a n b eobtained from the e igenval-u e s a n d c o r r e s p o n d i n geigenvectors of the load flowJacob ian. Using the functions in the toolbox and MAT-LAB, it is s t ra ight forward to s et u p a MATLAB scr ip t file(M-file) that performs a number of power f lows, eachw i t h i n c r e a s ed s y s t e m l o a d a n d g e n e r a t i o n . A t e a c hstage, th e load flow Jacobian is available, and its eigen-values an d eigenvectors can b e obtained us ing s tand ardMATLAB functions. In this example, the eigenvalues oft he inve rse of the Jacobian ar e determined. At vol tageinstability, the large st eigenvalue of th e inverse Jacobiantend s to infinity. A p ositive eigenvalue indicates a s t ab le

7I

Figure 3. Power system stabilizer14 IEEE ComputerApplications n Power


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sys tem; a negative eigenvalue indicates a n u nstable sys-t em. Th e e igenvec tor ind ica tes which of t h e s y s te mbuses take pa rt in the voltage instability.The change in th e cr i tical e igenvalue for the 68 bustes t case supplied with the toolbox, as he load and gen-eration ar e increased, is shown in Table 1.The b us vol t -age m agnitude profile in the po wer flow for this case isshow n in Figure 1.The eigenvector corresponding to thec r i t i ca l e igenva lue for 1 .3615 t ime s nomina l l oad i sshown in Figure 2. It can be seen that only a few of t h esys tem s buses are of significance to this mode. It canalso b e s e e n t h a t t h o s e b u s e s f o r w h i ch t h e c r i t ic a le igenvector has high values a re c losely re la ted t o lowbus voltages in th e power flow. The bu s with the largesteigenvector magnitude in this case correspo nds with thebus having the lowest voltage in the load flow. However,this may not be generally true. In this example, the eigen-value of th e Jacobian remains q ui te smal l unt i l a loadvery close to the maximum p ower level. Never-theless,the cr it ical mode sh apes are similar, even at th e nom inalsys t e m load . By inc rement ing the load s lowly in ther e g io n w h e r e t h e e i ge n v al u e s t a r t s t o i n c r e a se m o r equickly, i t is possible to approach closely to th e maxi-mum power point for the sys tem.Other exercises that could be performed by s tudentsinclude:Determine the best locations for capacitive suppo rtDetermine the v ol tage sens i t ivi ty to capaci ta t ive

Compare the ac t ion of addi t ional capaci tors andExamine the effect of gene rator var limits.

suppor ts ta t ic var com pensators

Power System Stabilizer DesignAnother app lication of modal analysis applied to powersystem stability is the determ ination of sys tem osc i l l a t ey instability and t he d esign of controls (power sys temstabi lizers) to e nsu re that th e sys tem is s table . In thisexample, the oscillations associated with a power gener-ation plant consisting of four identical gen erat ors , con -n e c t e d t h r o u g h t r an s f o r m e r s t o a c o m m o n b u s a r es tudied. The commo n bus is connected through a trans-mission line to a large gen erator that models th e rest ofthe interconnected sys tem. The sys tem m odel is shownin Figure 3a, and the excitation system and power sys-tem stab ilizer block diagram s in Figure 3b.Without a power sys t em s t ab i li ze r , t h e osc i l l a torymo de in which the plant oscillates against the rest of t h esys tem (the plant mode) is uns table . The other o sci l la te

within the plant. These mo des are stable. The power sys-tem stabilizer is used to s tabi l ize the plant m ode withoutcaus ing the intergenerator modes t o becom e uns table .Because the gene rators and their loading are as sumed tobe ident ical, there a re three sets of equal intergeneratormodes . A plot of the eigenvalues with no power system

d ry modes are associated with intergenerator oscillations

stabilizer is shown in Figure 4.Th e stabilizer design requires th e frequency respo nseof the aggregate generator model electrical torque to as i gn a l a p p l i e d t o t h e v o l t a g e r e f e re n c e p o i n t of t h eexciter. Th e aggregate model s obtained by parallelingthe generator t ransformer impedances and connect ing asingle generator and control model to t he LT bus havingfour times the rating of the individual generators. In thedetermin ation of the stabilizer s p ha se lead characteris-t i c , t h e i n e r t i a of t h i s a g g r e g a t e m o d e l s h o u l d b eincreased t o effectively eliminate chang es in spee d an dangle . The phase lag in the frequency response must becompensated by th e ph ase lead of th e power sys tem s ta-bilizer for ro bust stabilizer action. Th e ideal pha se lead

Ble Edit Mndmvs HelpSystem Eigenvalueswith no Stabilizer

t4 t 1

Figure 4. System eigenvalues with no stabil izerX = plant against system; = intergeneratorJanuary 1995 15


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and the p hase lead chosen for th e s tabil izer are show n inFigure 5. This is arrived at i teratively by re peatedly com -Table 2. Plant and inter-generatorelectromechanicalmodes

Ksfab= Wab=15 I stab-600.41 492j4.566 0.720292j4.4.3667 2.46042j2.3795-0.455462j6.9453 -2.7089ij9.2733 8.12772jl7.549

stabilizer frequency response1

70605040

i d e a l p h a s e l e a d

2010

0.5 1 1.5 2Figure5. Ideal and actual stabilizer phase leadcharacteristic

oo t Locus of System EigenvaluesII

I real partFigure 6. Plant and intergenerator eigenvalue locus a sstabilizer gain varies from 0 to 60 X = plant againstsystem; intergenerator

paring stabilizer frequency response with the precalcu-lated ideal phase comp ensation.Th e next s tage in stabilizer design s to determine theg a i n of t h e s t a b i l iz e r . T h i s is o f t en a c o m p r o m i s ebetween increasing the damping of the plant m ode with-out causing instability of oth er system modes. It is d o n eby repeated eigenvalue calculations on the original sys-tem model with th e stabilizer gain increasing. The locusof the plant and intergen erator mo des a s the s tabi lizergain increases is shown in Figure 6.Th e eigenvalues cor-responding to th e sys tem elect romechanical m odes wi thstabilizer gains of 0, 15,and 60 are provided in Table 2. Inthis example, little deterioration of intergenerator modedamping occurs and a gain of 15 is chosen as the recom-mend ed setting: higher gains may lead to noise prob lemsin the very high gain excitation system.Other exe rc i s es which could be pe r form ed by s tu-den ts include:Determinat ion of sens i t ivi ty of s y s t e m m o d e s ofoscillation to network, gene rator, and co ntrol para-metersExamination of interare a oscillations and th eir con-t ro l us ing power sys t em s t ab il i ze rs o r s t a t i c va rcompensators .AcknowledgmentFor Further ReadingMATLAB is a registered trademark of The Mathworks Inc.

J.H. Chow, K.W. Cheung, Toolbox for Power System Dynamics andControl Engineering Education and Research, IEEE Transactionson Power Systems November 1992, pages 1559-1564.P. Kundur, M. Klein, G.J. Rogers, M.S. Zywno, Application ofPower System Stabilizers for Enhancement of Overall System Stabil-ity, EEE Transactions on PowerSystem s May 1989, pages 614-621.B.Gao, G.K. Morison, P. Kundur, Voltage Stability AnalysisUsing Modal Analysis, IEEE Transactions on Power Systems,November 1992, pages 1529-1542.

MATLAB users manual, The Mathworks Inc., Natick, 1994.About the AuthorsGraham Rogers has had a varied career in power system engi-neering spanning over 40 years. After an engineering apprentice-shi p and serv ice in th e Royal Air Force, he entere d SouthamptonUniversity, UK, and graduated with first class honors in electricalenginee ring in 1961. After working as a consultant mathematicianat AEI Rugby) Ltd., he returned to Southampton University,wher e he taught until 1978. From 1978 to 1993, he was employedby Ontario Hydro, where h e worked in a special studies unit anddeveloped computer programs for power system dynamic analy-sis. On his retirement from Ontario Hydro, he formed CherryTree Scientific Software, which provides engineering services andmarkets power system software. He is also an associate professorpart-time) at McMaster University and an adjunct ass ociate pro-fessor at t he University of T oronto .

Joe Chow received his PhD degree in electrical engineeringfrom the University of Illinois at Urbana in 1977. From 1978 to 1987,he worked for the General Electric Company, Schenectady, NewYork. He is currently professor of electrical, computer , and sys-tems engineering and electric power engineering at RensselaerPolytechnic Institute. His technical activities include the develop-ment of engineering and educational tools for analyzing power sys-tem dynamics and performing control design.

6 IEEE Computer Applications in Power

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Hands on Teaching