Reliability-Based Methods for Electric System Decision Making

Transcription

1 Relablty-Based Methods for Electrc System Decson Makng Patrck Hester Old Domnon Unversty Engneerng Management and Systems Engneerng Department Kaufman Hall, Room 241 orfolk, VA Keywords: decson-makng, electrc system, RBDO, FORM, uncertanty Abstract The purpose of ths paper s to develop a methodology that utlzes relablty-based optmzaton to solve complex electrcal grd usage problems. Wth electrcal power grds, as wth many complex systems, complcated decsons must be made at both the local (user) and global (electrcty provder) levels; all decson makers have ndependent, often conflctng, obectves, further complcatng the decsons. In order to ncorporate both levels of decson makng (and resultng nteracton effects between the decson makers), a relablty-based optmzaton approach can be utlzed whch ncorporates local decson makers preferences by enforcng probablstc constrants on the overall optmzaton problem (e.g., sectors A and B need a partcular amount of power and each sector has a dfferent crtcalty level). Ths ensures that the optmzed decsons made at the global level satsfy the basc requrements of the local decson makers (e.g., to delver power to crtcal sectors). The uncertanty n ths approach s ncorporated through an effcent frst order relablty method (FORM), an analytcal approxmaton to falure probablty calculaton, rather than tradtonal, computatonally expensve smulaton-based methods (such as Monte Carlo samplng). Usefulness of ths methodology s shown through several example problems. 1. ITRODUCTIO As demand ncreases natonwde for electrcal power, the naton must look for ntellgent approaches to managng electrcal dstrbuton. Ths requres the development of an electrcty management approach that determnes the optmal allocaton of power to subsystems such that the cost of power s mnmzed. In order to acheve ths, complex electrcal grd usage problems requre the nteracton of ndvduals at both local and global levels. At the local level, users expect power to be avalable on demand (.e. wth 1% relablty). At the global level, electrcty provders are strugglng to meet the demands of ther customers n the most cost effcent manner possble. Thus, t benefts the electrcty provders to have the mnmum amount of power avalable so as not to waste electrcty when t s not beng used by customers. These obectves are nherently competng as mantanng electrcal servce avalablty for users s costly. Addtonally, the crtcalty of some nfrastructures (e.g. hosptals, polce statons) requres greater certanty of power avalablty than the average user. Combnng these factors results n a complcated decson makng problem. Ths paper develops a methodology for electrc system decson-makng at both a local and global level. It begns by dscussng a basc problem formulaton for local and global decson makng to facltate an nteroperable electrc grd. The detaled mathematcal approach behnd ths methodology s then dscussed. Sample problems employng ths methodology are demonstrated. Fnally some conclusons and recommendatons for problem extensons are dscussed. 2. PROBLEM FORMULATIO The goal of ths methodology s to develop an approach for electrc system usage whch ncorporates the needs of both local and global decson makers. The approach taken to acheve ths goal s to make decsons at a global level whch satsfy the constrants of local users. A global decson maker refers to the electrcty provder, and t can be a natonal power company or a cty-wde power company, for example. A local decson maker, on the other hand, s an electrcty user, and can nclude an entre cty s consumpton or a partcular sector of socety (such as a hosptal). For the remander of ths dscusson, electrc system decsons wll take the form of adustng the power output of generators whch, by vrtue of ther physcal connectons, can provde power to varous sectors (users) of socety. Fgure 1 shows an example electrc grd n whch ths type of decson makng may be necessary for provdng power to a hosptal, polce department and fre department. The global decson maker (n ths case, a cty s utlty company) must make a decson to provde power va any of Grd-Interop Forum 27 Paper_Id-1

2 the four generators to the three sectors. Whle t may seem obvous at frst to operate only generator 2 (snce t provdes power to all sectors), further analyss may ndcate that a combnaton of power to the other three generators may be optmal. Ths s due to the fact that one of the three sectors may be seen as more crtcal. Ths crtcal sector may requre a greater certanty of power avalablty. order to obtan a soluton that all nvolved decson makers fnd acceptable. Ths formulaton assumes complete certanty wth regards to power provded and power demanded. Ths s not accurate n the context of a real world applcaton. Therefore, the followng formulaton extends Eq. (1) to nclude uncertanty n PP and PD : Generator 1 Generator 2 Generator 4 mn Cost C d s.t. P PP d PD P crt (2) Hosptal FD Fre department Polce Department FD PD Generator 3 Fgure 1: Electrc Grd Illustraton In order to make electrc system decsons as descrbed above, ths paper proposes the followng problem formulaton: mn Cost s.t. PP d C d PD where s the ndex of generators n the power system, C s the cost assocated wth the th generator, d s a decson varable ndcatng the power level assocated wth the th generator (defned on [,1] where ndcates the generator s off and 1 ndcates the generator s operatng at 1% capacty), s the ndex of sectors n the power system, PP s the power provded by the th generator to the th sector, and PD s the total power demand of the th element n the system. In Eq. (1), the obectve s to mnmze the global decson maker s (n ths case, the electrc company s) overall cost of power generaton, gven constrants on the requred power mposed by sector users. Ths formulaton ensures that both local and global demands are beng met. Whle the optmal decson of Eq. (1) s not globally optmal (as the global decson maker s optmal cost s $ and the local decson makers would prefer to have all generators delverng power to ther sector at 1%), t s a soluton whch s satsfcng to all the nvolved decson makers. That s to say, the decson makers regard the soluton as good enough whle recognzng that t s not optmal for ther own self nterests [16]. Ths concept s essental when dealng wth complex, nteroperable systems. Sacrfces must always be made n (1) where P crt s the crtcalty probablty assocated wth the th sector, and all else s as before. Addtonally, the constrant whch refers to the power demand vs. the provded power s now defned probablstcally. That s, the net power must be delvered the th sector wth a probablty of at least P. Cost s assumed to be crt determnstc for the purposes of ths formulaton. Ths problem formulaton s smlar to the relablty-based desgn optmzaton problem formulaton, whch s dscussed n the followng secton. 3. RELIABILITY-BASED DESIG OPTIMIZATIO Relablty-based desgn optmzaton (RBDO) s concerned wth fndng a set of desgn varables for a gven engneerng system such that a gven obectve functon (mnmzaton of cost) s optmzed and the desgn requrements (power demand) are satsfed wth hgh probablty. As mentoned earler, the problem formulaton for RBDO s the same as n Eq. (2). Wthn the probablstc constrant,, whch s generally denoted as g ( ) PP d PD and s referred to as a performance functon n the RBDO lterature, s formulated such that g < ndcates falure, g > ndcates success, and g =, the boundary between falure and success s referred to as the lmt state. There are two steps n solvng Eq. (2). Step 1 s relablty analyss,.e., evaluaton of the probablty constrant. Step 2 s optmzaton. Step 1 s dscussed n detal below, focusng on a frst-order approxmaton to calculate the probablstc constrant n Eq. (2). Methods under step 2 are revewed later n ths secton. Step1: Analytcal calculaton of P(g ) requres the evaluaton of the ntegral of the ont probablty densty functon (pdf) of all the random varables over the falure doman, as P ( g ( d, x) ) f ( x) dx (3) x g ( d, x) Grd-Interop Forum 27 Paper_Id-2

3 where d s the set of decson varables and x s the set of random varables. Ths ntegral poses computatonal hurdles snce t can be dffcult to formulate the ont probablty densty explctly and ntegraton of a multdmensonal ntegral may be dffcult. Therefore, numercal ntegraton methods such as Monte Carlo smulaton or analytcal approxmatons such as frst-order relablty method (FORM) or second-order relablty method (SORM) are commonly used n mechancal systems relablty analyss. Monte Carlo smulaton requres multple runs of the determnstc system analyss and can be very costly. On the other hand, analytcal approxmatons such as FORM and SORM are very effcent, and have been shown to provde reasonably accurate estmates of the probablty ntegral for numerous applcatons n mechancal and structural systems. Detaled descrptons of these methods and computatonal ssues are provded n [1, 5, and 7]. In FORM, the varables, x, whch may each be of a dfferent probablty dstrbuton, and may be correlated, are frst translated to equvalent standard normal varables u. For uncorrelated normal varables, ths transformaton s smply u = x. (Later, ths concept s expanded to nclude varables that are non-normal and/or correlated). The lmt state and the falure and safe regons are shown n Fg. 2, n the equvalent uncorrelated standard normal space u. u2 g(u) < (falure) u* (mnmum dstance) g(u) = g(u) > (safety) u1 Fgure 2: Illustraton of lmt state and falure and safe regons The falure probablty s now the ntegral of the ont normal pdf over the falure regon. The FORM replaces the nonlnear boundary g = wth a lnear approxmaton, at the closest pont to the orgn, and calculates the falure probablty as P F = P( g (d,x) ) = (- ) (4) where P F s the falure probablty, s the cumulatve dstrbuton functon (CDF) of a standard normal varable and s the mnmum dstance from the orgn to the th lmt state. Thus, the multdmensonal ntegral n Eq. (3) s now approxmated wth a sngle dmensonal ntegral as n Eq. (4), the argument of whch (.e., ) s calculated from a mnmum dstance search. The mnmum dstance pont u* on the lmt state s also referred to as the most probable pont (MPP), snce lnear approxmaton at ths pont gves the hghest estmate of the falure probablty as opposed to lnearzaton at any other pont on the lmt state. (A secondorder approxmaton of the falure boundary s referred to as SORM, where the falure probablty calculaton also requres curvatures of the lmt state). The mnmum dstance pont (or MPP) u* s found as the soluton to the problem: mn (5) s.t. g (d,x) A ewton-based method to solve Eq. (5) was suggested by Rackwtz and Fessler [13]. Other methods such as sequental quadratc programmng (SQP) have also been used n the lterature [6] and [19]. For non-normal varables, the transformaton to uncorrelated standard normal space s u = x x, where x and are the equvalent normal mean and standard devaton, respectvely, of the x varables at each teraton durng the mnmum dstance search. Rackwtz and Fessler [13] suggested the soluton of x and by matchng the PDF and CDF of the orgnal varable and the equvalent normal varable at the teraton pont. Other transformatons are also avalable n [2, 11, 12, and 14]. If the varables are correlated, then the equvalent standard normals are also correlated. In that case, these are transformed to an uncorrelated space through an orthonormal transformaton of the correlaton matrx of the random varables through egenvector analyss or a Cholesky factorzaton [7]. The mnmum dstance search and frst-order or second-order approxmaton to the probablty ntegral s then carred out n the uncorrelated standard normal space. The mnmum dstance search typcally nvolves fve to ten evaluatons of the lmt state (and thus system analyss), and then the probablty s evaluated usng a smple analytcal formula as n Eq. (4). Compared to ths, Monte Carlo smulaton may need thousands of samples f the falure probablty s small, thus makng Monte Carlo methods prohbtvely expensve for solvng large scale stochastc optmzaton problems. Snce the lmt state functons nvolved n ths problem formulaton are lnear n the random varables, and the x Grd-Interop Forum 27 Paper_Id-3

4 random varables are assumed to be normal, FORM wll be accurate. Second order estmates [3, 8, and 18] of the falure probablty can also be used when the lmt state s nonlnear, but due to the smplcty of the lmt state functon n ths paper, second order methods are not found to be necessary. The mnmum dstance pont may also be found usng a dual formulaton of Eq. (5) as wth the equvalent KKT condton at the mnmum dstance pont on the lmt state. Several versons of decoupled and sngle loops have been developed, based on whether drect or nverse FORM s used for the relablty analyss step. ote that FORM s the key to all these effcent RBDO technques. Further nformaton on the use of FORM n varous RBDO formulatons can be found n [4]. mn g (d,x) s.t. u = crt Ths dual problem may be referred to as nverse FORM, and s used n the optmzaton (step 2) n ths paper. In ths formulaton, crt s set to value correspondng to P crt, as crt = 1 P ). ( crt Fgure 3: Illustraton of ested RBDO Method Step 2: In many mplementatons of relablty-based optmzaton, the probablty constrant n Eq. (2) s usually replaced by a quantle equvalent,.e., by a mnmum dstance constrant, as mncost( d) s.t. crt (7) where s the mnmum dstance computed from Eq. (5). Alternatvely, the dual formulaton has also been used, based on Eq. (6), as mncost( d) s.t. g ( d, x) where g (d,x) s computed from Eq. (6). Snce the relablty constrant evaluaton tself s an teratve procedure, the number of functon evaluatons requred for relablty-based optmzaton s consderably larger than determnstc optmzaton. A smple nested mplementaton of RBDO (.e., relablty analyss teratons nested wthn optmzaton teratons, as n Fgure 3) tremendously ncreases the computatonal effort, and as a result, several approaches have been developed to mprove the computatonal effcency, typcally measured n terms of the number of functon evaluatons requred to reach a soluton. In decoupled methods [6, 15, and 19], the relablty analyss teratons and the optmzaton teratons are executed sequentally, nstead of n a nested manner (refer to Fgure 4, where OL means optmzaton loop and RL means relablty loop). Ths s done by fxng the results of one analyss whle performng the teratons of the other analyss. Sngle loop methods [9, 1, and 17] perform the optmzaton through an equvalent determnstc formulaton whch replaces the relablty analyss constrant (8) Fgure 4: Illustraton of Decoupled RBDO wth Inverse FORM For the purpose of cost optmzaton n ths paper, a decoupled method of RBDO developed n [6] s chosen for reducng the number of functon evaluatons. In the cost optmzaton, a determnstc optmzaton s performed whch determnes a startng pont for the relablty analyss. Ths determnstc optmzaton follows the formulaton n Eq. (1) wth the mean values of the desgn varables servng as the values for the random varables. Ths optmzaton yelds a set of generator settngs (d s). These settng are then passed to the relablty analyss. The relablty analyss then calculates the power demand and power provded (x s or most probable ponts, known as the MPPs) n the orgnal varable space for the ndvdual lmt states, as outlned n Eq. (8). The MPP values are then fed back to the determnstc optmzaton and these values replace the random varable values (taken as the varable mean values n the frst teraton). Ths process contnues untl convergence s reached on both a confguraton and the MPP values. Ths process s outlned n Fgure 5 below. Grd-Interop Forum 27 Paper_Id-4

5 the varables), and optmum and nteger-only stochastc varables. The generator settng results from the example problems are shown below: Fgure 5: Decoupled Optmzaton Flowchart The cost optmzaton s performed va the branch and bound method and the relablty analyss s performed va the SQP algorthm. The next secton demonstrates ths methodology on an example problem. 4. EXAMPLE PROBLEM The example problem presented llustrates the problem formulaton developed n Sectons 2 and 3. The problem corresponds to the llustraton shown n Fgure 1. The generator data are as follows: x Cost PP Hosptal Fre Polce 1 25 (5,2.5) (5,.25) (1,.5) (5,.25) (5,.25) (15,.75) 4 2 (1,.5) - (2,1) Table 1: Generator Data The demand data for the sectors are as follows: Sector PD P crt Hosptal (5,2.5).9 Fre (1,.5).75 Polce (1,.5).75 Table 2: Demand Data It should be noted that all random varables are normally dstrbuted wth a coeffcent of varaton (COV = / ) of 5%. The electrc grd generator settngs were optmzed usng four confguratons: optmum (where generator settngs could take on any value between zero and one) and ntegeronly determnstc varables (evaluated at the mean values of Stochastc Determnstc Integer Integer Generator Optmum Only Optmum Only Table 3: Generator Settng Results The cost results from the example problems are shown below n Table 4. Cost Stochastc Optmum Stochastc Integer 5. Determnstc Optmum 39. Determnstc Integer 5. Table 4: Cost Results It s obvous that the nteger-only solutons are more expensve than the equvalent optmum solutons. Ths s due to the fact that nteger solutons represent power confguratons that are provdng the sectors wth excess power. Addtonally, t makes sense that the optmum value for the stochastc scenaro costs more money (.e. requres more power) than ts equvalent determnstc scenaro. Ths s because excess power must be provded to ensure that the requred demand s met wth the specfed P crt. The stochastc and determnstc nteger solutons result n the same settngs because they both provde a level of excess power that s adequate n both the determnstc and stochastc scenaros. 5. COCLUSIOS Utlzng a frst order relablty method, ths paper developed an effcent methodology for makng power system decsons at the global level that ncorporates the needs of local power system users. Ths methodology ncludes consderaton of uncertanty n power demand and provded power. Several extensons should be explored n future power system decson makng methodologes. They nclude: onlnear power functons. Delvered power and demand requre more complcated modelng than s present n ths methodology. Whle ths methodology s an approprate startng pont, realstc models should be ncorporated usng RBDO. on-lnear and non-determnstc costs. Cost s assumed to be both lnear (ncreasng as d ncreases) Grd-Interop Forum 27 Paper_Id-5

6 and determnstc. The effects of both non-lnear and stochastc costs should be nvestgated. More complcated sector nteractons. Sectors n ths paper do not have a drect nfluence on one another as they would n realstc scenaros (.e. as one sector s powered, the other has decreased power delvered). These nteractons should be nvestgated further and a more complcated nteracton model should be developed. References [1] Ang, A.H.-S. and Tang, W.H. (1984). Probablty Concepts n Engneerng Plannng and Desgn, Vol. II: Decson, Rsk, and Relablty, Wley, ew York. [2] Box, G.E.P. and Cox, D.R. (1964). An Analyss of Transformaton. Journal of the Royal Statstcal Socety, Seres B, 26, [3] Bretung, K. (1984). Asymptotc Approxmatons for Multnormal Integrals. Journal of Engneerng Mechancs, ASCE, 11(3), [4] Chralaksanakul, A. and S. Mahadevan. (25). "Frst- Order Methods for Relablty-Based Optmzaton." ASME Journal of Mechancal Desgn, 127(5): [5] Dtlevsen, O. and Madsen, H. O. (1996). Structural Relablty Methods. Wley, Y. [6] Du X., Chen W. (24). Sequental Optmzaton and Relablty Assessment method for effcent probablstc desgn. Journal of Mechancal Desgn, 126(2): [7] Haldar, A., and Mahadevan, S. (2). Probablty, Relablty, and Statstcal Methods In Engneerng Desgn. Wley, [8] Köylüoğlu, H.U. and elsen, S.R.K. (1988). ew Approxmatons for SORM Integrals. Structural Safety, 5, [9] Kuschel,. and Rackwtz, R. (2). A ew Approach for Structural Optmzaton of Seres Systems. In Melchers and Stewart (Eds.), Applcatons of Statstcs and Probablty ( ). Balkema, Rotterdam. [1] Lang J., Mourelatos Z. P., Tu J. (24). A Sngle Loop Method for Relablty-Based Desgn Optmzaton, Proceedngs of ASME Desgn Engneerng Techncal Conferences and Computer and Informaton n Engneerng Conferences, Salt Lake Cty, Utah, DETC24/DAC [11] Lu, P.L. and Der Kureghan, A. (1986). Multvarate dstrbuton models wth prescrbed margnals and covarances. Probablstc Engneerng Mechancs, 1(2): [12] ataf, A. (1962). Détermnaton des Dstrbuton don t les Marges Sont Donées. Comptes Rendus de l Academc des Scences, 225, [13] Rackwtz, R. and Fessler, B. (1978). Structural relablty under combned random load sequences, Computers and Structures, 9, [14] Rosenblatt, M. (1952). Remarks on a multvarate transformaton. Ann. Math. Stat., 23(3), [15] Royset, J.O., Der Kureghan, A., Polak, E. (21). Relablty-Based Optmal Structural Desgn by the Decoupled Approach, Relablty and Structural Safety, 73, [16] Smon, H.A. (1957). Models of Man: Socal and Ratonal. ew York: John Wley and Sons. [17] Tu J, Cho K. K., Park Y.H. (21). Desgn potental method for robust system parameter desgn. AIAA Journal, 39(4), [18] Tvedt, L. (199). Dstrbuton of Quadratc Forms n ormal Space Applcaton to Structural Relablty. Journal of Engneerng Mechancs, 116(6), [19] Zou T. and Mahadevan, S. (26). A drect decouplng approach for effcent relablty-based desgn optmzaton, Structural and Multdscplnary Optmzaton, 31(3), Bography Dr. Patrck T. Hester s an Assstant Professor of Engneerng Management and Systems Engneerng at Old Domnon Unversty. He receved a Ph.D. n Rsk and Relablty Engneerng (27) at Vanderblt Unversty and a B.S. n aval Archtecture and Marne Engneerng (21) from the Webb Insttute. Pror to onng the faculty at Old Domnon Unversty, he was a Graduate Student Researcher n the Securty Systems Analyss Department at Sanda atonal Laboratores, where he worked on developng a methodology for safeguard resource allocaton to defend crtcal nfrastructures aganst a multple adversary threat. Pror to that, he was a Proect Engneer at atonal Steel and Shpbuldng Company n charge of Modelng and Smulaton experments to test shpboard logstcs and cargo load-out capabltes. Hs research nterests nclude mult-obectve decson makng under uncertanty, resource allocaton, crtcal nfrastructure protecton, and network flows. Grd-Interop Forum 27 Paper_Id-6