INCORPORATING FLEXIBILITY IN THE DESIGN OF REPAIRABLE SYSTEMS DESIGN OF MICROGRIDS

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1 Proceedngs of IDETC/CIE 2014 ASME 2014 Internatonal Desgn Engneerng Techncal Conferences & Computers and Informaton n Engneerng Conference Buffalo NY, USA DETC INCORPORATING FLEXIBILITY IN THE DESIGN OF REPAIRABLE SYSTEMS DESIGN OF MICROGRIDS Vjtashwa Pandey 1 Annette Skowronska 1,2 pandey2@oakland.edu annette.g.skowronska.cv@mal.ml Zssmos P. Mourelatos 1 Davd Gorsch 2 Matthew Castaner 2 mourelat@oakland.edu davd.j.gorsch.cv@mal.ml matthew.p.castaner.cv@mal.ml 1 Mechancal Engneerng Department, Oakland Unversty, Rochester MI U.S. Army, TARDEC, Warren MI ABSTRACT The authors have recently proposed a decson-based framework to desgn and mantan reparable systems. In ther approach, a multobjectve optmzaton problem s solved to dentfy the best desgn usng multple short and long-term statstcal performance metrcs. The desgn soluton consders the ntal desgn, the system mantenance throughout the plannng horzon, and the protocol to operate the system. Analyss and optmzaton of complex systems such as a mcrogrd s however, computatonally ntensve. The problem s exacerbated f we must ncorporate flexblty n terms of allowng the mcrogrd archtecture and ts runnng protocol to change wth tme. To reduce the computatonal effort, ths paper proposes an approach that learns the workng characterstcs of the mcrogrd and quantfes the stochastc processes of the total load and total supply usng autoregressve tme-seres. Ths allows us to extrapolate the mcrogrd operaton n tme and elmnate therefore, the need to perform a full system smulaton for the entre long-term plannng horzon. The approach can be appled to any reparable system. We show that buldng n flexblty n the desgn of reparable systems s computatonally feasble and leads to better desgns. 1. INTRODUCTION Many tmes relable power s needed n remote locatons, where ether a utlty connecton s not avalable or t s unrelable. In contrast to a sngle power source, mcrogrds are systems of nterconnected sources and loads managed usng ntellgent power control software (Crawford, 2013). A mcrogrd s slanded and most lkely part of an emergency operaton f t s located n an area wthout a major cty utlty supply (Skowronska et al., 2013). The mltary can use slanded mcrogrds n war zones, where the mplcatons of relablty and cost can be crtcal. Whle many defntons of relablty exst (Kapur and Lamberson, 1977), we consder a mcrogrd relable f t supports the loads wth mnmum servce nterruptons. Ths relablty comes however, at a monetary and logstcal cost. In ths paper, we desgn a mcrogrd archtecture consderng relablty and cost measures by treatng t as a reparable system whose operaton s characterzed usng a Mnmal Set of Metrcs (Pandey et al., 2013). A tme-seres modelng approach s used to characterze the load and supply random processes (.e., learn ther statstcal behavor through tme) usng short-term data and then use them to nfer the mcrogrd operaton n the long run. We also ncorporate flexblty n desgn by allowng the mcrogrd to respond to changes n operatng condtons by changng ts runnng protocol and archtecture n order to reman optmal throughout ts operatonal tme UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release Copyrght 2014 ASME Internatonal

2 Report Documentaton Page Form Approved OMB No Publc reportng burden for the collecton of nformaton s estmated to average 1 hour per response, ncludng the tme for revewng nstructons, searchng exstng data sources, gatherng and mantanng the data needed, and completng and revewng the collecton of nformaton. Send comments regardng ths burden estmate or any other aspect of ths collecton of nformaton, ncludng suggestons for reducng ths burden, to Washngton Headquarters Servces, Drectorate for Informaton Operatons and Reports, 1215 Jefferson Davs Hghway, Sute 1204, Arlngton VA Respondents should be aware that notwthstandng any other provson of law, no person shall be subject to a penalty for falng to comply wth a collecton of nformaton f t does not dsplay a currently vald OMB control number. 1. REPORT DATE 22 APR REPORT TYPE Journal Artcle 3. DATES COVERED to TITLE AND SUBTITLE Incorporatng Flexblty n the Desgn of Reparable Systems - Desgn of Mcrogrds 5a. CONTRACT NUMBER W56HZV b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) Vjtashwa Pandey; Annette Skowronska; Zssmos Mourelatos; Davd Gorsch; Matthew Castaner 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Mechancal Engneerng Department,Oakland Unversty,2200 N. Squrrel Road,Rochester,M, SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) U.S. Army TARDEC, 6501 East Eleven Mle Rd, Warren, M, d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 8. PERFORMING ORGANIZATION REPORT NUMBER ; # SPONSOR/MONITOR S ACRONYM(S) TARDEC 11. SPONSOR/MONITOR S REPORT NUMBER(S) # DISTRIBUTION/AVAILABILITY STATEMENT Approved for publc release; dstrbuton unlmted 13. SUPPLEMENTARY NOTES For Proceedngs of IDETC/CIE 2014 ASME 2014 Internatonal Desgn Engneerng Techncal Conferences & Computers and Informaton n Engneerng Conference 14. ABSTRACT The authors have recently proposed a "decson-based" framework to desgn and mantan reparable systems. In ther approach, a multobjectve optmzaton problem s solved to dentfy the best desgn usng multple short and long-term statstcal performance metrcs. The desgn soluton consders the ntal desgn, the system mantenance throughout the plannng horzon, and the protocol to operate the system. Analyss and optmzaton of complex systems such as a mcrogrd s however, computatonally ntensve. The problem s exacerbated f we must ncorporate flexblty n terms of allowng the mcrogrd archtecture and ts runnng protocol to change wth tme. To reduce the computatonal effort, ths paper proposes an approach that "learns" the workng characterstcs of the mcrogrd and quantfes the stochastc processes of the total load and total supply usng autoregressve tme-seres. Ths allows us to extrapolate the mcrogrd operaton n tme and elmnate therefore, the need to perform a full system smulaton for the entre long-term plannng horzon. The approach can be appled to any reparable system. We show that buldng n flexblty n the desgn of reparable systems s computatonally feasble and leads to better desgns. 15. SUBJECT TERMS

3 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unclassfed b. ABSTRACT unclassfed c. THIS PAGE unclassfed Publc Release 18. NUMBER OF PAGES 9 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescrbed by ANSI Std Z39-18

4 (plannng horzon). We argue that mcrogrds provde a representatve test-bed n the desgn and analyss of reparable systems. For reparable systems (Crow, 1974; Rgdon and Basu, 2000; Wang and Cot, 2005), the classcal noton of relablty s not drectly applcable. The classcal relablty, defned as the probablty that the system has not faled before a gven tme t, may be msleadng because the system may have faled and repared before tme t (Haldar and Mahadevan, 1999). The classcal relablty defnton also mpedes decson makng nvolvng mantenance, avalablty and servce cost of reparable systems. A good mantenance strategy whch makes a system avalable most of the tme cannot compensate for too many servce nterruptons and a potentally hgh servce cost. The tradeoffs between performance, servce nterruptons and cost are hard to capture. Pandey and Mourelatos (2013) have recently shown that we can systematcally approach the desgn and mantenance of reparable systems usng the followng three-steps: 1. Identfcaton of the objectves (metrcs) for optmal operaton as elcted from the decson maker, 2. Modelng of the reparable system over tme, ncludng up-tme and down-tme perods, and 3. Smultaneous optmzaton of the dentfed objectves over a pre-specfed plannng horzon. In the frst step, the decson maker usually dentfes pertnent performance requrements whch are then converted nto mathematcal metrcs from whch a Mnmal Set Of Metrcs (MSOM) s obtaned. A MSOM addresses all system requrements and therefore, captures most of the nformaton about the operatonal condtons and reparablty of the system. Step 1: Metrc Identfcaton The reparable system n ths paper s a remote electrc (smart chargng) mcrogrd. The system optmzaton problem nvolves multple conflctng objectves addressng: 1. One or more measures of relablty such as perods of unnterrupted operaton or number of falure and repar events, 2. Mnmzaton of captal and runnng cost, and 3. A measure of longevty; e.g., the plannng horzon. These consderatons gude the selecton of the MSOM as outlned later n the paper. Step 2: Mcrogrd Modelng The power-load characterstcs of a mcrogrd are managed by turnng sources and loads on and off based on how much a load or source s above or below set ponts. These set ponts are not known a pror and are usually desgn varables n the optmzaton process. Other requrements such as the number of sources and loads as well as the nventory are also consdered. The mcrogrd smulaton over a long tme horzon (e.g., one year) may be acheved by ether smulatng over a short tme duraton (e.g., hours) and extrapolatng over the long tme horzon, or by usng a coarse tme scale over the long tme horzon. Unless done properly, what s learned n a short smulaton tme may not be applcable to the entre plannng horzon. If a coarse tme scale s used to cover a long plannng horzon, many transent effects can be mssed snce they happen wthn seconds. Transent effects may for example, nclude the mbalance caused when generators and nvertors react dfferently to a change n load condtons. A smulaton usng large tme steps may mss these effects. On the other hand, usng a fne tme scale s computatonally prohbtve for long smulatons. Another major challenge n smulatng complex reparable systems, and n partcular mcrogrds, s the effect of uncertanty. Unless models wth a fne tme scale are used over long perods of tme, we cannot obtan relable nformaton about the system. For example, we may encounter a chance falure and assume that the mcrogrd s unrelable or alternatvely by luck, we may not see any falure n a short perod even f the mcrogrd s unrelable. To properly capture the effect of uncertanty, multple replcatons of the system response are requred to estmate the statstcs of performance metrcs. Ths ncreases the computatonal effort even more. Step 3: Multobjectve Optmzaton A mathematcal optmzaton over the dscussed attrbutes s generally set up as a non-lnear mxed nteger problem because of the non-lnear objectve functons and the combnaton of dscrete and contnuous decson varables (Whtefoot et al., 2011). In our work we use a multobjectve genetc algorthm (GA). 1.1 Flexble desgn and our approach Our dscusson so far ndcates that the optmal desgn of a mcrogrd s computatonally challengng. The need to analyze mcrogrds accurately and n detal stems from the need for flexble operaton. A flexble mcrogrd responds to changes n operatng condtons by changng ts runnng protocol and archtecture n order to reman optmal throughout the operatonal tme UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release Copyrght 2014 ASME Internatonal

5 perod. In essence, a flexble mcrogrd avods a sngle pont desgn, an approach used n stochastc optmzaton, where a robust desgn wth the hghest expected utlty s chosen (de Neufvlle and Scholtes, 2011). Flexblty allows the system to change tself accommodatng uncertan events and ncreasng therefore, ts relablty. To address the tme scale and length of analyss ssues, we propose usng tme-seres modelng to learn the characterstcs of the load and supply random processes from a short operaton, and extrapolate thereafter to a longer perod. The runnng protocol of a mcrogrd nfluences ts need to servce a total load by supplyng a total power. The learnng we propose, characterzes the load random process and determnes the characterstcs of the supply random process n response to the load. Ths ensures that the relatve levels of load and supply and ther correlaton n tme are captured. Once the load and supply processes are quantfed, we can use them to estmate long term relablty metrcs. Ths approach s theoretcally sound as opposed to extrapolatng the results of a short smulaton run. As we have mentoned, short smulatons are prone to mssng extreme events and therefore, relatve frequency of falure events are lkely to be mscalculated. Modelng the statstcs of the entre process elmnates ths ssue. A reparable system must be desgned over multple metrcs that capture dfferent facets of the system performance. We wll show how to calculate the metrcs of the system usng a quantfed random process approach. Our method can be used to also assess new concepts such as grdable vehcles. When vehcle systems lnk nto a mcrogrd n order to share power sources and loads, a vehcle-to-grd (V2G) system s formed (Kempton and Tomc, 2005). The analyss and desgn of such mcrogrds s complcated and a method to extrapolate what s learned n a short tme perod can be very useful. The paper s organzed as follows. Secton 2 descrbes our methodology ncludng the chosen mnmal set of metrcs and the tme-seres modelng of the load and supply stochastc process. Secton 3 provdes detals on the desgn and operaton of a smart chargng mcrogrd ncludng results and a senstvty analyss. Fnally, Secton 4 concludes and dscusses future research drectons. 2. METHODOLOGY 2.1 Metrc-based desgn of reparable systems The classcal relablty theory of reparable systems uses metrcs such as the Mean Tme Between Falures (MTBF) and avalablty to characterze the performance of a reparable system (Haldar and Mahadevan, 1999). These metrcs are calculated usng tmes between falures and system repar frequency and duratons. However, they only capture one statstc of the tme to falure. The MTBF captures the mean, whle the avalablty s smply the rato of system up-tme to the total duraton the system s n operaton. A system that has a skewed dstrbuton of the tme between falures wll not have ts performance well represented by the MTBF or avalablty. To account for these lmtatons, we have proposed usng a Mnmal Set of Metrcs (MSOM) to fully descrbe the performance of reparable systems (Pandey and Mourelatos, 2013). The MSOM should be defned so that the metrcs, ndvdually or collectvely, cover most aspects of the system performance. The reader s referred to (Pandey and Mourelatos, 2013) for more detals. For the mcrogrd example of ths paper, we use the metrcs of cost (C), mnmum falure free perod wth 80% probablty (T 0.8 ), and number of falures (N f ) wthn the plannng horzon. In the optmzaton phase, a Pareto front s generated over these metrcs, whch are then traded off by the decson maker to obtan the optmal combnaton that best satsfes hs/her preferences. 2.2 Mcrogrd operaton Mcrogrds consst of many nterconnected loads and sources and nclude ntellgent power management to enable a robust and relable operaton wth substantal fuel and mantenance economes over ther servce lfe. Most mcrogrds have an AC module and a DC module. The AC module connects and dsconnects dfferent sources and loads and handles the AC power. The DC module manages the DC sources (e.g., batteres and solar arrays), and nverts them to get AC power whch s then suppled to the AC module. Common power sources n a mcrogrd nclude a utlty man f avalable, generators, solar arrays, wnd turbnes and rechargeable vehcles. The sources are gven prorty numbers to determne the reverse order n whch they wll be taken offlne, f necessary. A low number ndcates that the source s crtcal and wll be taken offlne after the other sources have already been taken offlne. The mcrogrd load s also explctly modeled. The sources and loads are shed and added dependng on the system s excess capacty. Common loads nclude buldngs, battery chargng, and other mscellaneous loads. Each load has also a prorty number. At each tme, the runnng protocol of the mcrogrd decdes, based on load requrements, how many sources to keep on, what capacty to run them at, and how much excess capacty to have n order to account for stochastc load varatons. If we characterze the load random process and therefore, know ts behavor through tme, UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release Copyrght 2014 ASME Internatonal

6 Power we can account for sudden load changes by sutably ncreasng or decreasng the supply. However, because of nherent uncertanty n load and partal system falures, ths s not always possble. Falures are due to nadequate capacty to servce the load and/or to actual subsystem falures. Falures are expected because cost consderatons usually preclude a mcrogrd from never falng. For a dynamc mcrogrd system, t s very dffcult to predct ts performance n the long term wthout smulatng the grd for a long duraton. Consderng that most electrcal transents happen n mllseconds, we must run smulatons wth a very short tme step for months at a tme n order to fully account for them. Ths s of course computatonally mpractcal. We propose to learn the characterstcs of the load profle L(t) and the resultng supply profle S(t), as enacted by an ntellgent power management protocol. It s necessary to not only characterze the random processes L(t) and S(t) but also the correlaton between them. Ths s because a supply that s not well correlated wth load wll ether lead to falures, waste of power because of overproducton when not needed, or both. A short perod of a few days can be used for the learnng process. Based on the quantfed stochastc behavor, we can then extrapolate the two random processes for a long tme (several months or even years) and record the number of falures where the supply s less than the load (Fg. 1) and the tmes falure occurred. Ths nformaton can be used to quantfy the system performance metrcs. Falure Tme Supply, S(t) Load, L(t) Fgure 1. Realzatons of the supply and load random processes combnaton of the two to determne future observatons. AR models are the most commonly used. They use a weghted average of past observatons n addton to a whte nose error term, to capture the correlaton at tmes t and t 2 where t2 t1 s small. 1 t Consder a random process X t. A sample functon x s dscretzed n the tme nterval [0, T] usng a unform tme step t so that x xt and t 1 t. For the AR(p) model, the dscretzed sample functon s represented as where x 1( x1 ) 2( x2 ( x p p ) )... (1a) s the temporal mean of the process, 2 N(0, e ) s Gaussan whte nose and 1, 2,... p, are feedback parameters to be estmated. The parameter p ndcates the order of the autoregressve process whch n turn, ndcates the level of correlaton to past observatons. Eq. (1a) can be used to create a dervatve process as x y L (1b) where s mean of the process Y(t) at tme, x s a zero mean process modeled usng the AR model of Eq. (1a), and L s a functon of x. To preserve statonarty and ergodcty of x, L must be lnear and constant. After the feedback parameters are estmated n Eq. E t X t Xˆ t s formed as (1a), a resdual seres the dfference between the actual estmated X t and the Xˆ t processes and statstcal tests are performed to make sure the random varables E t and E t are uncorrelated for every. Detals are provded n (Ruppert, 2004). 2.3 Tme seres modelng of load and supply We use tme seres modelng to characterze the load and supply random processes. Tme seres models have been extensvely used to characterze random processes (Ruppert, 2004; Nkolads et al., 2011). They combne Auto-Regressve (AR) models, Integrated (I) models, and a Movng Average (MA) model. The Integrated (I) model s used f the process s non-statonary. All models use a feedback mechansm based on ether past observatons, past standard errors (MA), or a UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release 3. A SMART CHARGING MICROGRID EXAMPLE As dscussed n Secton 2, the mcrogrd mplements control by sensng power usage at varous loads and routng power to and from several components to brng the system to the desred state of operaton. Ths entals swtchng contactors on or off. When ntated, the grd starts at the system equlbrum and remans n ths state unless/untl the excess system capacty moves outsde specfed set-ponts. Excess capacty s defned as the Copyrght 2014 ASME Internatonal

7 avalable power n excess of the current load. The mcrogrd s assumed faled f t cannot meet the load requrements;.e., the nstantaneous supply S(t) s less than the nstantaneous load L(t). There are varous scenaros where ths can happen such as the total capacty s not enough to meet an unexpected spke n load, one or more sources or contactors have faled, a software error n mplementng the control has occurred, or any combnaton of the above. If a falure occurs, we ether repar the faled component(s) or wat for the mcrogrd to recover f the supply capacty s reached (soft falure). One way to obtan the optmal archtecture of a mcrogrd (Pandey et al., 2013), s by solvng the multobjectve optmzaton problem of Eq. (2). For that, we use the Non-domnated Sortng Genetc Algorthm II (NSGA-II) (Deb et al., 2002). The three objectves of cost (C), mnmum falure free perod wth 80% probablty (T 0.8 ), and number of falures (N f ) wthn the plannng horzon are consdered. The desgn varables are the set-ponts where the loads and sources are taken offlne/onlne, and the number of sources (generators, solar arrays and contactors) we start wth. Mn T N f, C x 0.8, where: x s, s, s, s, n, n, n ls T subject to: so 0.8 lo F : P 8760 g x n, n gen ss 1 T workng gen 0.2 C C ntal C repar solar, n contacts, n batt s, s, s, s, 0,100 ls so lo ss N solar T contacts (2) The smulaton s run for 8760 hours (1 year) usng a coarse tme scale for each value of the desgn varable vector. One of the solutons, shown n Table 1, from the generated Pareto front s chosen for further analyss. Fg. 2 shows the source and load behavor for about 90 hours of operaton for the soluton n Table 1. At certan tme nstances the load s hgher than the supply because of the stochastcty n the load and/or component falures. Ths s expected because no system can be fully relable. Fgure 2. Realzatons of load and supply processes for the optmal soluton n Table 1 Table 1. One of the Pareto front solutons Decson Value varables s 0.6% ls s 18% so s 11% lo s 22% ss n 6 gen n 17 contacts Attrbutes T 563 hrs 0.8 N 6 C f $1.08M Smulatng the mcrogrd operaton through tme, even for specfed values of the desgn varables, requres a substantal computatonal effort because for each tme step we must obtan detals of the mcrogrd operatonal state such as sensng the load, turnng sources and loads off/on and reparng/replacng components as needed. To address ths ssue, we propose that the mcrogrd be smulated for only a short perod of tme (much shorter than the one year plannng horzon) and use the smulated results to characterze the supply and load processes. Ths s possble only f the stochastc part of the processes after subtractng the trend s statonary. In other words, we propose to use short tme data to learn the statstcal characterstcs of the process for the entre plannng horzon. Subsequently, realzatons of the characterzed processes can be used to calculate the attrbutes n Eq. (2) wthout carryng out smulatons. To demonstrate our approach, we assume that the mcrogrd load s a dervatve process as n Eq. (1b). The UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release Copyrght 2014 ASME Internatonal

8 random part s represented by the zero-mean, fourthorder, AR model of Eq. (3) where tme t s measured n hours. 2t Lt sn + 20( (3) ) The random process of Eq. (3) captures pertnent aspects of loads encountered n real lfe. The process has a trend wth a stable element of 50 kw, and a snusodal element wth perod of a day and a 40 kw ampltude. The latter smulates durnal load changes. The stochastc part has four feedback parameters and a whte nose term represented by the standard normal random varable. The standard devaton of the whte nose at each tme step s = 6.89 kw. In realty, the tme seres model s determned from actual load realzatons observed over a short perod of tme. Snce the supply s a functon of load, we model t usng the same fourth-order AR model of Eq. (3) wth two changes. Frst, the whte nose n the supply s correlated wth the whte nose n the load and second, there s an excess capacty bult-n to ensure that the supply s n general, greater than the load. In prncple, a process representng the tme varyng dfference between the load and supply may be used. It s preferable however, to model the load and supply processes separately because we have no control over the load and the supply s controlled by the runnng protocol of the mcrogrd. Two strateges are consdered for mplementng the excess capacty. In the frst strategy, we multply the determnstc part of the load by a factor of 1, and n the second strategy we add a fxed addtonal power to the modeled load. Eqs (4) and (5) express the two strateges. 2t St sn + 20( ) (4) 2t St sn + 20( (5) -2-3 The correlaton between and s denoted by whch s less than one. The hgher ts value, the better the mcrogrd control algorthm knows the value of the load ncludng the uncertan component of t. A UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release -4 value of 1 mples that the load stochastcty s perfectly modeled by the algorthm and thus, a supply value to meet the load can be theoretcally created wth no excess capacty. In ths case, the mcrogrd wll never fal unless some of ts components fal. We vary the value of to nvestgate ts effect on the mcrogrd performance metrcs. Fg. 3 shows one realzaton of the load and supply processes accordng to Eqs (3) and (4) for 0. 9 and 11. We note that the supply s generally hgher than the load, except for some nfrequent chance falures because of the load stochastcty. Fgure 3. Realzatons of smulated load and supply processes for Strategy 2 We now evaluate dfferent workng scenaros of the mcrogrd whch s optmzed over the three long term metrcs of cost, MFFP and number of falures. Strategy 1 uses a percentage of the supply as excess supply and Strategy 2 generates a fxed addtonal (excess) power over the assessed load. To compare the two strateges, we use the cost of excess capacty C e. In our opnon, the cost of operaton s best measured by the cost of excess capacty the mcrogrd generates over tme. Ths s expected to be less than the cost of settng up or runnng the mcrogrd. Because t s mperatve to meet the requred loads, the cost to do so cannot be consdered a good performance metrc. A mcrogrd can be expensve f t must meet hgher load requrements, and cannot be called worse than a mcrogrd that s cheaper only because the load requrements are low. The best mcrogrd therefore, s the one that best allocates the excess power to mnmze falures. For ths example, the cost to generate power s 10 cents per kwh. The number of falures N f s the number of dstnct tmes the mcrogrd fals durng the plannng horzon of one year. Recall that the proposed method can be used to analyze a grd for many years wth a very small addtonal computatonal effort. The thrd metrc of MFFP, T 0.8, s calculated usng the runnng duratons of Copyrght 2014 ASME Internatonal

9 the grd between falures. It s smply the 20 th percentle of the runnng duratons. overall cost of excess capacty wll reman relatvely constant. 3.1 Senstvty analyses Senstvty analyses are presented here for dfferent model parameters and optmal solutons for Strategy 2. We show that Strategy 2 provdes a better way to buldn excess capacty. We frst compare the two strateges to buld excess capacty accordng to Eqs (4) and (5). Realzatons of the mcrogrd load and supply random processes are generated for 8760 hours. For comparson purposes, we calculated the values of and n Eqs (4) and (5) that result n the same number of falures (equal to 6) durng the plannng horzon. The soluton of Table 1 generates 135,002.4 kwh of excess energy, costng $13, (135,002.4 kwh tmes 10 cents per kwh) n nsurance aganst chance falures. Fxng the number of falures at 6, Strategy 1 generates 329, kwh of excess energy over the course of the year for a $32, cost of nsurance aganst chance falures. In contrast, Strategy 2 wth an excess power of 11 kw generates only 96, kwh of extra energy for only a cost of $9, Strategy 2 s therefore, the most cost effectve and we use t for further analyss. The takeaway s that buldng extra supply capacty as a percentage of the load s wasteful because when the load s hgh, there s less lkelhood of t ncreasng substantally any more whle the opposte s true when the load s low. Therefore, we should buld extra capacty n terms of fxed power n kw and not as a percentage of the load. Next we study the senstvty of the number of falures and of the cost wth respect to the correlaton coeffcent between the load and supply nose terms (Eqs 3 and 5). Fg. 4 shows the expected decrease n the number of falures. The falures fall to zero when the supply s perfectly correlated wth the load. Whle ths seems ntutve, there are two thngs to notce. Frst, accurate modelng of the load, and a mcrogrd response wth a supply that meets that load quckly, are essental for a relable mcrogrd. The shorter the delay n the response, the better the correlaton between load and supply would be. Second, the cost does not change much wth the value of the correlaton coeffcent. Ths s because when the supply s equal to the sum of the expected load and a fxed load, we can drectly calculate the surplus generaton (and hence cost) over a year by multplyng the number of hours a mcrogrd s operatonal by. Ths number s not affected much by the actual value of the load f the stochastc element s small compared to the mean (see Eq. 3). The change n cost due to error n assessng the load because of a low value of wll also average out over tme and the UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release Fgure 4. Effect of ncreasng correlaton between load and sources on number of falures We now concentrate on the optmal soluton usng Strategy 2. We wll determne the optmal extra power (excess capacty) n Eq. (5). Snce the problem nvolves multple attrbutes we expect to get multple non-domnated solutons. Fg. 5 shows the Pareto front over the two attrbutes of cost and number of falures. The decson maker can use t to choose the most desrable combnaton of attrbutes based on hs/her preferences. The optmal depends on the selected cost and the number of falures from the Pareto front. Recall that there are three attrbutes. For smplcty, the attrbute of MFFP s not shown. If the decson maker selects the desgn wth an approxmate cost of $8,842 and 20 falures, the soluton s to select Strategy 2 to provde excess supply and use a value of 10 kw for. The correspondng MFFP value wll be hours. Of course, ths s contngent on beng equal to 0.9. Whle quantfyng the load and source processes, we must ensure that they and the correlaton between them are modeled correctly. The proper choce of supply as a functon of tme, ncludng the optmal excess capacty, depends on how accurately the load s modeled. Copyrght 2014 ASME Internatonal

10 Fgure 5. Tradeoff between cost from excess capacty and number of falures for Strategy 2 Our results show that we can characterze the load and supply random processes usng short duraton smulatons (.e., learn the operatonal characterstcs of the mcrogrd) and use them to predct the stochastc behavor of load and supply for long duraton smulatons. In ths case, further analyss of operatng scenaros, ncludng the optmal strategy of addng excess capacty, s possble wth lmted computatonal effort. For sudden changes of the load profle, the correspondng supply profle can be establshed relatvely easly. However to obtan the mcrogrd archtecture n response to these changes, whch may ental addng or removng generators or electrc vehcles, wll requre more sophstcated models. One approach s to buld a Krgng model (Sacks et al., 1989) between the mcrogrd archtecture and the optmal supply profle. A truly flexble desgn can then be realzed where rapdly changng needs are met quckly and optmally wth mnmal computatonal effort. 4. CONCLUSIONS AND FUTURE WORK Reparable systems are hard to analyze because many classcal relablty engneerng concepts do not apply readly. Mcrogrds are reparable systems that can be excellent test-beds n research, desgn and analyss of reparable systems. Remote systems have become more electrfed over tme and ther power needs and generaton capabltes have ncreased. Ther optmal operaton s therefore, crtcal. A mcrogrd as a collecton of power sources for a specalzed use must be relable (meetng ts ntended purpose under uncertanty most of the tme) over a long tme perod at a reasonable cost. Avalable optmzaton methods usng longduraton smulatons may not be practcal because of hgh computatonal effort. We must smulate the mcrogrd operaton for a long perod of tme n order to properly quantfy the tme-dependent load and the requred supply as mplemented by the chosen protocol. UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release Many smulatons are necessary to obtan the optmal mcrogrd desgn that provdes the rght trade-off between dfferent performance metrcs under uncertanty. Because the smulatons are computatonally expensve, many shortcuts are used n the lterature such as usng a coarse tme scale gnorng therefore transents, or runnng the smulaton for a short tme to the detrment of properly estmatng long term performance metrcs. Ths paper addressed ths mportant ssue usng a dfferent approach. We proposed to quantfy the load and supply random processes usng actual load and supply data over a short tme and establsh the correlaton between the two processes. The quantfed processes are then used to generate load and supply realzatons over the long plannng horzon and calculate performance metrcs of nterest such as cost, number of falures and MFFP. Because actual smulatons are performed over a short tme, or for a lmted number of desgns, the computatonal cost s mnmal. We showed that ths s a feasble strategy to optmze mcrogrds over varous metrcs and also perform senstvty analyss on varous model assumptons. Usng our method, we made two recommendatons on how to desgn a mcrogrd. Frst, we should model the load stochastc process as accurately as possble, thereby ncreasng the chance that supply wll be hgher than the load most of the tme durng the tme of nterest. Second, an optmal supply of power can be obtaned by addng a fxed and optmally determned amount of power, to the value of the stochastc load at any tme nstead of a percentage of the load. Not only does t lead to hgher relablty but t also makes the calculaton of runnng cost easer. In future work, we wll quantfy the load and supply random processes to nclude dscrete chance events of falure of components or ntentonal shuttng off of a load or a source. Furthermore, we wll reduce the computatonal effort usng tme-dependent metamodels between the mcrogrd archtecture and ts long-term operaton ncludng measures of relablty and operatonal cost. ACKNOWLEDGMENT We would lke to acknowledge the techncal and fnancal support of the Automotve Research Center (ARC) n accordance wth Cooperatve Agreement W56HZV U.S. Army Tank Automotve Research, Development and Engneerng Center (TARDEC) Warren, MI. REFERENCES 1. Crawford, M., 2013, Mcrogrds May Power Entre U.S. n Future, Copyrght 2014 ASME Internatonal

11 topcs/artcles/renewable-energy/mcrogrds-maypower-entre-future Date accessed, 24th Jan Crow, L. H., 1974, Relablty Analyss for Complex, Reparable Systems n Relablty and Bometry, eds F. Proschan & R.J. Serfng, SIAM, , Phladelpha. 3. De Neufvlle and Scholtes, S., 2011, Flexblty n Engneerng Desgn, MIT Press, Cambrdge MA, p Deb, K., Pratap, A., Agrawal, S. and Meyarvan, T., 2002, A Fast Eltst Non-domnated Sortng Genetc Algorthm for Mult-objectve Optmzaton: NSGA-II, IEEE Transactons on Evolutonary Computaton, 6(2), Haldar, A. and Mahadevan, 1999, Probablty Relablty and Statstcal Methods n Engneerng Desgn, 1st Edton, John Wley and Sons. 6. Kapur, K. C. and Lamberson, L. R., 1977, Relablty n Engneerng Desgn, 1 st Edton, John Wley and Sons. 7. Kempton, W. and Tomc, J., 2005, Vehcle-to-grd Fundamentals: Calculatng Capacty and Net Revenue, Journal of Power Sources, 144(1), Nkolads, E., Mourelatos, Z. P. and Pandey, V., 2011, Desgn Decsons under Uncertanty wth Lmted Informaton, 1 st Edton, Taylor and Francs. 9. Skowronska, A., Gorsch, D., Pandey, V., Mourelatos, Z. P., Mange, J. and Dunn, A., 2013, Global Strateges for Optmzng the Relablty and Performance of a US Army Moble Power Transfer System, Proceedngs of the 2013 Ground Vehcle Systems Engneerng and Technology Symposum (GVSETS), Troy, MI. 10. Pandey, V., Skowronska, A. G., Mourelatos, Z. P., Gorsch, D. and Castaner, M., 2013, Relablty and Functonalty of Reparable Systems usng a Mnmal Set of Metrcs: Desgn and Mantenance of a Smart Chargng Mcrogrd, ASME Internatonal Desgn Engneerng Techncal Conferences, Paper DETC , Portland, OR. 11. Pandey, V. and Mourelatos, Z. P., 2013, New Metrcs to Assess Relablty and Functonalty of Reparable Systems, SAE Internatonal Journal of Materals and Manufacturng, 6(3), Rgdon, S. and Basu, A., 2000, Statstcal Methods for the Relablty of Reparable Systems 1 st Edton, Wley-Interscence, pp Ruppert, D., Statstcs n Fnance, Sprnger Texts, Sacks, J., Welch, W. J., Mtchell, J. J., and Wynn, H. P., 1989, Desgn and Analyss of Computer Experments, Statstcal Scence, 4(4), Wang, P. and Cot, D. W., 2005, Reparable Systems Relablty Trend Tests and Evaluaton Proceedngs of the Annual Relablty and Mantanablty Symposum, Whtefoot, J., Mechtenberg, A. R., Peters, D. L. and Papalambros, P. Y., 2011, "Optmal Component Szng and Forward-lookng Dspatch of an Electrcal Mcrogrd for Energy Storage Plannng," Proceedngs of the ASME 2011 Internatonal Desgn Engneerng Techncal Conferences, Washngton, DC. UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release Copyrght 2014 ASME Internatonal