Multiyear and Multi-Criteria AC Transmission Expansion Planning Model Considering Reliability and Investment Costs

Transcription

1 Multiyear Multi-Criteria AC Transmissin Expansin Planning Mdel Cnsidering Reliability Investment Csts Phillipe Vilaça mes, Jã Pedr ilva Jã Tmé araiva INEC TEC FEUP/EEC ept. Eng. Eletrtécnica e de Cmputadres da Faculdade de Engenharia da Universidade d Prt - Prtugal phillipe.gmes@fe.up.pt, japedrsilva91@gmail.cm, jsaraiva@fe.up.pt Abstract One f the majr cncerns in Pwer ystems is surely related with their reliability. Lng-term expansin planning studies traditinally use the well-knwn deterministic N-1 cntingency criterin. Hwever, this criterin is applied based n wrst-case analyses the btained plan may riginate ver-investments. ifferently, prbabilistic reliability appraches can incrprate different type f uncertainties that affect pwer systems. In this wrk, a lng term multi-criteria AC Transmissin Expansin Planning mdel was develped cnsidering tw bjectives - the prbabilistic reliability index Expected Energy Nt upplied (EEN) the investment cst. The Paret-Frnt assciated with these tw bjectives was btained using enetic Algrithms the final slutin was selected using a fuzzy decisin making functin. This apprach was applied t the IEEE 24 Bus Test ystem the results ensure its rbustness efficiency. Index Terms Multi-Criteria Multi-Year Transmissin Expansin Planning, Paret-Frnt, Fuzzy ecisin Making, EEN, AC-Optimal Pwer Flw. I. INTROUCTION The main bjective f the Transmissin Expansin Planning (TEP) Prblem is t define where, when hw a transmissin system shuld be mdified in rder t adequately meet the future dem. This exercise can be cnducted in rder t cnsider several bjective functins as minimizing the investment peratin csts, increasing the system reliability, minimizing the greenhuse gas emissins, increasing the flexibility f system peratin while reducing the netwrk charges, prviding a better vltage prfile, etc. Besides, the TEP prblem can be mdelled cnsidering static r a dynamic (multiyear) appraches. In the first ptin, the study is perfrmed cnsidering each perid at a time in a way that the equipments (transmissin lines, cables r transfrmers) selected t exp the system in ne perid are cnsidered n the basis tplgy fr the subsequent nes. On the ther h, the multiyear appraches take the hrizn in a hlistic way the prblem is slved in a single run fr all perids. It is ntewrthy that the multiyear TEP preserves the hlistic planning view this is essential t btain gd quality lng-term expansin plans. The experience f the authrs als indicates that slving the TEP prblem using a static year by year apprach riginates a glbal slutin with a cst that is nt inferir t the cst assciated t the multiyear apprach. This result is just the applicatin f the well-knwn rule indicating that the aggregatin f partial ptima is nt superir t the ptimum cming frm a glbal analysis. One f the majr cncerns in Pwer ystems is surely related with their reliability, which in turn can be studied using deterministic r prbabilistic appraches. Traditinally, TEP is cnducted using the well-knwn deterministic N-1 criterin. This criterin is applied based n wrst-case analyses (draw frm single cntingency). Hwever, this apprach des nt define cnsistently the true risk f the system, since it des nt take int accunt hw systems perates, hw cmpnents fail the existence f different lad levels. In additin, the N-1 criterin usually riginates ver-investments [1]. ifferently, prbabilistic appraches allw incrprating uncertainties assciated t the nn-ideal behavir f pwer system cmpnents. Meantime, the prbabilistic appraches require pre-defining the reliability indices t be used, which in turn, may intrduce sme subjectivity in the analysis. As stated in [2], these aspects explain the dichtmy between deterministic prbabilistic appraches regarding the TEP prblem. In this paper a Multi-Criteria Multi-Year Transmissin Expansin Planning was develped cnsidering tw bjectives - the investment cst the prbabilistic reliability index Expected Energy Nt-upplied (EEN). The develped apprach uses a Nn-minative CHA-Climbing enetic Algrithm (NCCA) t build the Paret-Frnt f the ptimizatin prblem using this frnt it is then used a Fuzzy ecisin Making Functin (FMF) t select the final expansin plan. Regarding the structure f the paper, fllwing this Intrductin, ectin II presents the AC mdel fr the TEP prblem ectin III prvides a brief analysis f Reliability studies in pwer systems as well as the main steps f the chrnlgical Mnte Carl imulatin t estimate the EEN index. ectin IV details the NCCA tl its main blcks, ectin V presents a brief descriptin f Fuzzy The research leading t this wrk is funded by the CAPE under BEX number

2 ecisin Making ectin VI prvides the results btained in the simulatins. Finally ectin VII includes sme cmments the cnclusins abut this wrk. II. TEP MATHEMATICAL FORMULATION The mst adequate mdel t deal with TEP prblems is the AC pwer flw (AC-OPF) peratin mdel because it cnsiders the reactive pwer, the lsses the bus vltage limits. Hwever, this mdel is mre deming frm a cmputatinal pint f view than C based mdels, thus requiring the use f very efficient ptimizatin techniques t slve AC-OPF prblems. The AC-OPF used in this paper is frmulated by (1) t (9). 2 Min COP = αi 1. Pi + αi2. Pi + αi3 (1) subject t PV (, θ, n) P + P = 0 (2) QV (, θ, n) Q + Q = 0 (3) P P P (4) min Q Q Q (5) min V V V () min frm ( N + N) ( N + N) (7) t ( N + N) ( N + N) (8) 0 n n (9) In this frmulatin P( V, θ, n) QV (, θ, n) are calculated by (10) (11), the bus cnductance susceptance B are given by (12) (13). PV (, θ, n) = Vi Vj[ ( n).cs θ + B ( n).sin θ ] (10) QV (, θ, n) = Vi Vj[ ( n).sin θ B ( n).cs θ ] (11) ( n) = ( n. g + n. g ) = ii ( n) = ( n. g + n. g ) j Ωi B ( n) = ( n. b + n. b ) B = sh sh sh Bii ( n) b [ n ( b b ) n ( b b )] = j Ωi frm The apparent flws calculated by (14) (15) where P, Q, t Q are given by (1) t (19). frm frm 2 frm 2 (12) (13) t in branch are frm frm t P = ( P ) + ( Q ) (14) t t 2 t 2 = ( P ) + ( Q ) (15) frm P 2 = Vi. g Vi. Vj ( g.cs θ + b.sin θ ) (1) frm 2 sh Q = Vi.( b + b) Vi. Vj( g.sin θ b.cs θ) (17) t P = V 2. g V. V ( g.cs θ b.sin θ ) (18) j i j t 2 sh Q = V.( b + b ) + V. V ( g.sin θ + b.cs θ ) (19) j i j In this frmulatin, the bjective functin (1) crrespnds t the peratin cst f a thermal system where α i1, α i2 αi3 are cefficients f the quadratic generatr cst functins f each generatin unit i dispatching a real pwer. P is the real pwer generatin, Q is the reactive pwer generatin, P is the real pwer dem, Q is the reactive pwer frm t dem, V is the vltage magnitude, are the branch apparent flws in terminals, g b are the cnductance the susceptance f branch i-j. III. RELIABILITY IN POWER YTEM A. Overview n pwer system reliability methds In the assessment f pwer systems reliability, there are tw different appraches that allw evaluating the adequacy f a pwer system that can be used in TEP mdels: deterministic prbabilistic appraches. eterministic criteria are based n a pre-specified rule that is defined cnsidering the experience btained thrugh the analyses f ther pwer systems. In TEP, the N-1 deterministic criterin t mdel likely failures is ften taken in accunt. Hwever, this apprach des nt cnsider the stchastic behaviur f pwer systems. This means that in rder t cnsider uncertainties related t pwer systems such as the pssible failure f system cmpnents, the weather cnditins r the dem grwth, a stchastic mdel shuld be fllwed. Markv prcesses are the well-knwn reference that allw including different system states. Figure 1 shws a typical tw-state Markv mdel in which the failure repair rates are mdelled by expnential distributins. In ther wrds, these distributins mdel the duratin f the system events. Figure 1: Markv mdel cmpsed by tw states, where λ is the failure rate μ is the repair rate. Regarding the prbabilistic appraches, there are tw main families f methds that shuld be mentined: the analytical the simulatin nes. The calculatin f system reliability indices is the main gal f bth appraches. Hwever the applicatin f analytical methds t cmplex pwer systems is nt adequate due t the number f simplificatins assumptins that usually need t be accepted. The simulatin prcesses are cmmnly knwn as Mnte Carl imulatins (MC). MC uses a rm sampling f states in rder t estimate the reliability indices. There are tw different types f MC: the chrnlgical the nn-chrnlgical. ince the develped TEP apprach uses the EEN t characterize the system reliability, it is imprtant t keep track f the sequence f states determining the life f the system tgether with their duratin. Therefre, in this paper, the chrnlgical MC will be used t estimate the EEN index as it will be detailed in the next ectin.

3 B. Chrnlgical Mnte Carl imulatin t estimate the Expected Energy Nt upplied, EEN As mentined befre, the develped multiyear TEP prblem includes the EEN t measure the reliability f the system under analysis. If just a nn-chrnlgical MC was used, each system state was characterized by the n r ff state f each cmpnent, an AC OPF culd be run cnsidering the n cmpnents the assciated Pwer Nt upplied culd then be btained. Hwever, there wuld be n indicatin abut the duratin f each state in the sense that the peratin-failure cycle f each cmpnent was nt cnsidered. In rder t get mre insight n system peratin t pass frm PN t EN it is necessary t sample peratin repair times f the cmpnents thus justifying the use f a chrnlgical Mnte Carl t estimate the EEN. The chrnlgical r sequential MC apprach requires using the failure density functin t mdel the peratin repair times f each cmpnent usually mdeled by an expnential distributin [3]. The main blcks f the chrnlgical MC t evaluate reliability indices are presented belw. Prcedure Chrnlgical MC Initialize system data: MTTF, MTTR, pre-specified β Initialize the system state: thrugh the selected prbability distributin, sample the perating time f each system cmpnent. Repeat Pick the lwest sampled time. Let F be the assciated cmpnent. Evaluate the system state. Accrding t this system state, calculate the pwer nt supplied then the energy nt supplied multiplying the pwer nt supplied by the duratin f the current system state. Update the accumulatrs f the reliability indices. epending n the previus state f cmpnent F, sample a new perating r repair time. Update the state the lifetime f F. Evaluate the system lifetime. Fr each year f simulatin, the reliability indices must be updated as well as the cefficient f variatin β. Until cefficient f variatin β r the imum number f years is reached. End Chrnlgical MC The previus sequential MC blcks shw that fr each system cmpnent an peratin-repair life cycle is develped. Using this cycle, the energy nt supplied can be calculated fr each system state cnsequently it is pssible t btain an estimate fr EEN. This index reflects states in which the lad is nt fully supplied because it exceeds the available generatin capacity /r because there is insufficient branch transmissin capacity. IV. NON-OMINATIVE HILL-CLIMBIN ENETIC ALORITHM, NCCA The main blcks f the NCCA are similar t the nes f a genetic algrithm applied t slve the TEP prblem. Additinally, it includes an imprvement ppulatin blck, a Tabu list t cntrl the diversity f the ppulatin a genetic similarity cntrl t ensure the elite diversity thrugh the generatins. The main NCCA blcks are presented belw detailed in the next sectins. Prcedure NCCA et the list f prjects having nprj elements. Initialize a rm ppulatin with ps individuals. Repeat Reprductin Mutatin Imprvement Evaluatin electin imilarity Cntrl tp Test Until test is psitive End NCCA A. Pssible prjects The earch pace Reductin The planner shuld specify a list f pssible prjects defined in terms f the extreme ndes, type (verhead line r transfrmer), transmissin capacity investment cst. The TEP algrithm shuld then select sme f the elements in this list t be integrated in the expansin plan lcate them in ne f the years f the hrizn. The current apprach restricts these prjects t crridrs already used in the base tplgy althugh it can be adapted t allw using new crridrs. On the ther h, using a large number f branches can turn the cmputatinal effrt t slve the TEP prblem prhibitive. Therefre, the search space was reduced using a Cnstructive Heuristic Algrithm (CHA) detailed in [4]. B. Particle Cdificatin Each individual crrespnds t an expansin slutin plan it is encded by a vectr that includes as many genes as the number f equipments in the reduced list cming frm the CHA mentined in IV.A. Each gene cntains an integer number that represents the perid f the planning hrizn in which this equipment will be inserted int the netwrk. C. Creating the Initial Ppulatin The initial ppulatin is rmly created with the aid f a Tabu List which ensures the diversity f this ppulatin.. Reprductin In the reprductin blck pairs f individuals rmly chsen are used t create an ffspring. ifferently frm usual crssver strategies in which just ne psitin in each

4 individual is sampled, in this case we sample tw psitins. tarting with the first individual we sample anther ne t frm a pair, then tw psitins are sampled in these tw individuals ne ffspring is created. This prcedure is repeated until all individuals underg reprductin thus creating a new ppulatin with the same size as the initial ne. E. Mutatin The mutatin nly affects a small percentage f the ffsprings it aims at increasing the diversity f a particular individual. Once an individual is selected, a particular gene is rmly sampled then its assciated integer number is mdified. If this integer number is increased, than this means pstpning the cnstructin f the assciated equipment. F. Imprvement Blck The imprvement blck is based n the individual's characteristics, that is, if an individual has an unacceptable value fr PN fr a particular year, it is imprved inserting new equipments selected using a CHA. On the ther h, if this individual displays a PN value belw a threshld, it is mdified by eliminating equipments using the Hill Climbing Methd. Accrding t this methd, an equipment is remved the new individual is evaluated again. If the resulting PN value cntinues belw the threshld this change is cnfirmed. If nt, that equipment is included back in the individual. These algrithms are detailed in [5] they are used as a way t accelerate the cnvergence f the genetic algrithm reducing the investment cst r the EEN index f sme slutins.. Evaluatin In the evaluatin blck the values f investment cst EEN are calculated. In a first mment, the PN is btained by running the AC-OPF using (1) t (9). If this value is greater than zer the fitness functin (20) is calculated the EEN is set at (high value). If the PN is zer the EEN is btained as described in ectin III. In (20) p is the perid in study, np is the ttal number f perids, r is the return rate is the penalty factr fr PN., = (1 + ) +. (20) H. electin The selectin is perfrmed in tw steps as fllws. In the first place, the nn-dminated slutins are selected, regarding the bjectives investment csts EEN, included in the new ppulatin. In the secnd step it is used a turnament selectin with the remaining slutins t cmplete the ppulatin until its riginal size is reached. I. enetic imilarity Cntrl The nn-dminated slutins cming frm the first step f the electin prcess are subjected t a similarity cntrl t increase the diversity f the new ppulatin. This cntrl uses the cncept f parent circle f radius r as illustrated in Figure 2. Fr each nn-dminated slutin a parental circle is used arund it. If anther slutin is lcated inside this circle, then the slutin having the highest investment cst is discarded. Figure 2: Illustratin f the cncept f parental circle. J. tpping Criterin The stpping criterin is based n the set f nn-dminant slutins. If this set remains unchanged fr a pre specified number f iteratins the iterative prcess ends. V. FUZZY ECIION MAKIN Once the Paret-frnt is built, the decisin maker has a number f slutins characterized by the crrespnding investment cst EEN. In rder t help the decisin maker t select the final slutin we used a fuzzy decisin making functin accrding t which the membership functin μi measures the adequacy f each slutin k in the Paret frnt regarding a specific bjective functin fr each i ( f1 = CINV f 2 = EEN ). Figure 3 illustrates the cncept assciated with this apprach. Fr each bjective functin i, fi, the decisin maker specifies a minimum a imum level, s that if a slutin k has a value fr f i less than the minimum, then a 1,0 membership degree is assigned. If the value f f i is between the minimum the imum then a membership degree decreasing frm 1,0 t 0,0 is assciated indicating that slutin k is less cmpatible with the cncept f being a gd quality slutin in the sense f minimizing the bjective f i. Finally, if the value f f i is larger than the imum level then the slutin k has bad quality it is eliminated. Once having the membership degrees regarding the tw bjectives fr a slutin k, then the minimum f the tw is taken this value is used t characterize slutin k. Amng all the slutins in the Paret Frnt, it will be selected the ne having the imum f the minimum values f the crrespnding bjectives as it is translated by (20). ecisin = k(min( μ( CINV ), μ( EENk))) (21) Figure 3: Fuzzy Mechanism fr Best Cmparisn VI. TET AN REULT The Nn-minative CHA-Climbing enetic Algrithm described in ectin IV was applied n the mdified IEEE 24 Bus Reliability Test ystem. The system used in the tests has sme differences regarding the riginal system prpsed in [] the system details can be fund in [3] [4]. k

5 The tests were perfrmed cnsidering the multiyear TEP mdel with 3 perids using a lad increase f 5% per perid. The riginal list f equipments includes 38 lines transfrmers in the first place the CHA was applied t select a sub-list f cidate equipments cnsidering a static TEP prblem fr each perid. After slving 23 AC-OPFs, using the MATPOWER tl described in [7], running in MATLAB with an Intel i7, 3.4Hz, 8 B RAM, the CHA selected 10 branches fr pssible reinfrcement frm the initial 38 equipments leading t a reductin f 99% in the search space. The equipments that result frm search space reductin step are the nes cnnecting buses 1-5, 3-24, -10, 7-8, 11-13, 13-23, 14-1, 15-24, 1-17, 17-18, in which it is permitted build up t 3 circuits fr each path. The simulatins were perfrmed in MATLAB with an Intel i7, 3.4Hz, 8 B RAM. The NCCA used 50 individuals in the ppulatin the parent circle radius was 4 set at 10. The parameters fr the CHA the Hill Climbing methds are the same as used in [5]. The simulatin invlved slving abut AC-OPFs in abut 100 hurs. Figure 4 shws the slutins btained, in which the vertical axis crrespnds t the present value f the investment cst, that is, the sum f the investment cst in the three perids brught back t the departing perid using a return rate f 5% per year. The hrizntal represents the EEN fr the three years. After building the Paret-Frnt, it was used the Fuzzy ecisin Making described in ectin V. Fr each Paret Frnt slutin k, it was calculated the membership functin given by (20). The final slutin crrespnds t slutin 3 in Figure 4 detailed in Table I. VII. CONCLUION This paper presents a dynamic apprach f the Transmissin Expansin Planning prblem using a multicriteria analysis that cnsiders the ttal investment cst the Expected Energy Nt upplied as the bjectives. The prblem was slved using the Nn-minative CHA- Climbing enetic Algrithm tl develped by Vilaça araiva in [5]. A Fuzzy decisin making prcess is then used t select the final expansin plan amng the slutins in the Paret Frnt. The EEN index was estimated using a Chrnlgical Mnte Carl imulatin. The NCCA was applied t the mdified IEEE 24-Bus Reliability Test ystem shwed excellent perfrmance t deal with the large cmputatinal effrt required t estimate the EEN t slve the AC-OPFs. The tl prvided a set f 4 final slutins with different values f investment cst EEN. On the ther h, the adptin f multi-bjective appraches is very relevant as a way t turn the TEP prblem mre realistic. It als ffers a trade-ff analysis between bjectives instead f ther techniques that, fr instance, require transfrming all bjectives except ne in cnstraints r building a value functin that aggregates all the individual bjectives using weights specified by the user. As future wrk, the develped tll can be parallelized t take advantage f multi-cre machines. ACKNOWLEMENT P. Vilaça acknwledges the financial supprt frm CAPE chlarship under Bex Figure 4: Paret-Frnt assciated with the investment cst EEN. The shape f this frnt deserves a cmment because it behaves differently regarding the theretical illustratin in Figure 3. The TEP prblem has a discrete nature s that including a new equipment in the expansin plan r changing the set f equipments t pass, fr instance, frm slutin 4 t slutin 3 leads t discrete jump in the investment cst. Apart frm that, the evaluatin f EEN requires slving nn-linear AC prblems. Tgether, these tw issues determine that a cnvex shape has the ne in Figure 3 is nt always btained in the case f the TEP prblem. TABLE I. BET OLUTION IENTIFIE BY THE FUZZY ECIION MAKIN Perid New equipment Invest. Cst ( ) EEN (MWh) 1 1-5, -10, 7-8, , , 11-13, (2) 14-1, , 3-24, , ,87.10 REFERENCE [1] J. P. V. Vasques, efinitin f maintenance plicies in pwer systems, M.c Thesis, epartment f Electrical Cmputer Engineering - University f Prt, [2] J. R. P. Barrs, A. C.. Mel, A. M. L. da ilva, An apprach t the explicit cnsideratin f unreliability csts in transmissin expansin planning, in 2004 IEEE 11th Internatinal Cnference n Prbabilistic Methds Applied t Pwer ystems, 2004, pp [3] R. Billintn, Reliability Evaluatin f Engineering ystems - Cncepts Techniques. New Yrk, [4] P. V. mes J. T. araiva, Hybrid iscrete Evlutinary PO fr AC ynamic Transmissin Expansin Planning, in IEEE Energycn cnference, 201. [5] P. V. mes J. T. araiva, Hybrid enetic Algrithm fr Multi-Objective Transmissin Expansin Planning, in IEEE Energycn cnference, 201. [] P. ubcmmittee, IEEE Reliability Test ystem, IEEE Trans. Pwer Appar. yst., vl. PA-98, n., pp , Nv [7] R.. Zimmerman, C. E. Murill-anchez, R. J. Thmas, MATPOWER: teady-tate Operatins, Planning, Analysis Tls fr Pwer ystems Research Educatin, IEEE Trans. Pwer yst., vl. 2, n. 1, pp , Feb