Linear Programming Makes Railway Networks Energy-efficient

Transcription

1 Linear Programming Makes Railway Neworks Energy-efficien arxiv: v1 [mah.oc] 27 Jun 2015 Shuvomoy Das Gupa, J. Kevin Tobin and Lacra Pavel Absrac In his paper we propose a novel wo-sage linear opimizaion problem o calculae energy-efficien imeables in elecric railway neworks. The resulan imeable minimizes he oal energy consumed by all rains and maximizes he uilizaion of regeneraive energy produced by braking rains, subec o he consrains in he railway nework. In conras o oher exising models, which are N P-hard, our model is compuaionally he mos racable one being a linear program. The model can be applied o any railway nework of arbirary opology using exising echnology. We apply our opimizaion model o differen insances of service PES2-SFM2 of line 8 of Shanghai Mero nework spanning a full service period of one day 18 hours) wih housands of acive rains. For every insance, our model finds an opimal imeable very quickly larges runime being less han 13s) wih significan reducion in effecive energy consumpion he wors case being 19.27%). Index Terms Railway neworks, energy efficiency, regeneraive braking, rain scheduling, linear programming. 1 Inroducion 1.1 Background and moivaion Efficien energy managemen of elecric vehicles using mahemaical opimizaion has gained a lo of aenion in recen years [1, 2, 3, 4, 5]. Calculaing energy-efficien imeables for rains in railway neworks is a relevan problem in his regard. Elecriciy is he main source of energy for rains in mos modern railway neworks; in such neworks, a rain is equipped wih a regeneraive braking mechanism ha allows i o produce elecrical energy during is braking phase. In his paper, we formulae a wo-sage linear opimizaion problem o obain an energy-efficien imeable for a modern railway nework. The imeable schedules This work was suppored by NSERC-CRD and Thales Canada Inc. S. D. Gupa and L. Pavel are wih he Edward S. Rogers Deparmen of Elecrical and Compuer Engineering, Universiy of Torono, and J. K. Tobin is wih Thales Canada Inc. shuvomoy.dasgupa@mail.uorono.ca, Kevin.Tobin@halesgroup.com, pavel@conrol.uorono.ca) 1

2 he arrival ime and he deparure ime of each rain o and from he plaforms i visis such ha he oal elecrical energy consumed is minimized and he uilizaion of produced regeneraive energy is maximized. 1.2 Relaed work The general imeabling problem in a railway nework has been sudied exensively over he pas hree decades [6]. However, very few resuls exis ha can calculae energy-efficien imeables. A Mixed Ineger Programming MIP) model, applicable only o single rain-lines, is proposed by Peña-Alcaraz e al. [7] o maximize he oal duraion of all possible synchronizaion processes beween all possible rain pairs. The model is hen applied successfully o line hree of he Madrid underground sysem. However, he model can have some drawbacks. Firs, considering all rain pairs in he obecive will resul in a compuaionally inracable problem even for a moderae sized railway nework. Second, for a rain pair in which he associaed rains are far apar from each oher, mos, if no all, of he regeneraive energy will be los due o he ransmission loss of he overhead conac line. Finally, he model assumes ha he duraions of braking and acceleraing phases say he same wih varying rip imes, which is no he case in realiy. The work in [8] proposes a more racable MIP model, applicable o any railway nework, by considering only rain pairs suiable for regeneraive energy ransfer. The opimizaion model is applied o he Dockland Ligh Railway and shows a significan increase in he oal duraion of he synchronizaion process. Alhough such increase, in principle, may increase he oal savings in regeneraive energy, he acual energy saving is no direcly addressed. Similar o [7], his model oo, assumes ha even if he rip ime changes, he duraion of he associaed braking and acceleraing say he same. Oher relevan works implemen mea-heurisics such as simulaed annealing [9] and geneic algorihm [10] and [11]. However, hese models can no be applied beyond small sized rain-lines. As he models use mea-heurisics, hey canno provide any guaranee of opimaliy of he calculaed imeable. Also, he qualiy of he resulan imeables is heavily dependen on uning of he parameers, which is always done manually and changes from one railway nework o anoher. An insighful analyical sudy of a simplified and periodic railway schedule appears in [12]. However he model lacks periodic even scheduling consrains [13], which are used o model cyclic imeables. As a resul, if he period is sricly smaller han he duraion of he railway service, which is ofen he case in periodic railway neworks [14, pages 7-10], hen he model canno be applied, because in such railway neworks rains associaed wih he nex period ener he nework while he rains associaed wih he firs period are sill running. 1.3 Conribuions Our conribuions in his paper are as follows: 2

3 We propose a novel wo-sage linear opimizaion problem o calculae an energyefficien railway imeable. The firs opimizaion model minimizes he oal energy consumed by all rains subec o he consrains presen in he railway nework. The problem can be formulaed as a linear program, wih he opimal value aained by an inegral vecor. Based on he soluion for he firs problem, he final opimizaion model maximizes he ransfer of regeneraive braking energy beween suiable rain pairs, while keeping he oal rain energy consumpion a he minimum. In conras o exising relevan models, all of which are of N P-hard complexiy, our opimizaion model is a linear program, hence is compuaionally he mos efficien one. We apply our model o eleven differen insances of service PES2-SFM2 of line 8 of Shanghai Mero nework. The imeables span full service period of one day 18 hours) wih each insance having housands of acive rains. For every insance, our model finds an opimal imeable very quickly wih he larges runime being 12.58s. In comparison wih he original imeables, he final imeables produced by our model reduces he oal effecive energy consumpion significanly, even he wors case reducion being 19.27%. To he bes of our knowledge, in comparison wih oher relevan models, ours is he only one ha calculaes energy-efficien railway imeable of such large scale in such a shor CPU ime. 1.4 Organizaion This paper is organized as follows. In Secion 2 we describe he noaion used, and hen in Secion 3 we model and usify he consrains presen in he railway nework. The preliminary opimizaion model is presened in Secion 4. Secion 5 formulaes he final opimizaion problem ha addiionally maximizes he uilizaion of regeneraive braking energy. In Secion 6 we apply our model o differen insances of an exising railway nework spanning a full working day and describe he resuls. Secion 7 presens he conclusion. 2 Noaion and noions Every se described in his paper is sricly ordered and finie unless oherwise specified. The cardinaliy and he ih elemen of such a se C is denoed by C and Ci) respecively. The se of real numbers and inegers are expressed by R and Z respecively; subscrips + and ++ aached wih eiher se denoe non-negaiviy and posiiviy of he elemens respecively. A column vecor wih all componens one is denoed by 1. The symbol sands for componenwise inequaliy beween wo vecors and he symbol sands for logical and. The number of nonzero componens of a vecor x is called cardinaliy of ha vecor and is denoed by cardx). The ih uni vecor e i is he vecor wih all componens zero 3

4 excep for he ih componen which is one. The epigraph of a funcion f : C R where C is any se) denoed by epi f is he se of inpu-oupu pairs ha f can achieve along wih anyhing above, i.e., epi f = {x, ) C R x C, fx)}. The convex hull of any se C, denoed by conv C, is he se conaining all convex combinaions of poins in C. Consequenly, if C is nonconvex, hen is bes convex ouer approximaion is conv C, as i is he smalles se conaining C. The se of all plaforms in a railway nework is indicaed by N. A direced arc beween wo disinc and non-opposie plaforms is called a rack. The se of all racks is represened by A. The direced graph of he railway nework is expressed by N, A). The se of all rains is denoed by T. The ses of all plaforms and all racks visied by a rain in chronological order are denoed by N N and A A respecively. The decision variables are he rain arrival and deparure imes, o and from he associaed plaforms, respecively. Le a i and d i be he arrival ime and he deparure ime of he rain T o and from he plaform i N. 3 Modelling he consrain se In his secion we describe he consrain se for our opimizaion model. This comprises he feasibiliy consrains for a railway nework of arbirary opology, and he domain of he decision variables. In mos of he exising railway neworks, he railway managemen has a feasible imeable; we use he sequence of he rains from ha imeable. The lower and upper bound of he consrains are inegers represening ime in seconds. 3.1 Trip ime consrain The rip ime consrains play he mos imporan role in rain energy consumpion and regeneraive energy producion. These can be of wo ypes as follows Trip ime consrain associaed wih a rack Consider he rip of any rain T from plaform i o plaform along he rack i, ) A. The rain depars from plaform i a ime d i, arrives a plaform a ime a, and i can have a rip ime beween τ and τ. The rip ime consrain can be wrien as follows: T i, ) A τ a d i τ ). 1) Trip ime consrain associaed wih a crossing-over A crossing-over is a special ype of direced arc ha connecs wo rain-lines, where a rainline is a direced pah wih he se of nodes represening non-opposie plaforms and he 4

5 se of arcs represening non-opposie racks. If afer arriving a he erminal plaform of a rain-line, a rain urns around by raversing he crossing-over and sars ravelling hrough anoher rain-line, hen he same physical rain is reaed and labelled funcionally as wo differen rains by he railway managemen [14, page 41]. Le ϕ be he se of all crossingovers, where urn-around evens occur. Consider any crossing-over i, ) ϕ, where he plaforms i and are siuaed on differen rain-lines. Le B be he se of all rain pairs involved in corresponding urn-around evens on he crossing-over i, ). Le, ) B. Train T urns around a plaform i by ravelling hrough he crossing-over i, ), and beginning from plaform sars raversing a differen rain-line as rain T \{}. A ime window [κ, κ ] has o be mainained beween he deparure of he rain from plaform i labelled as rain ) and arrival a plaform labelled as rain ). We can wrie his consrain as follows: 3.2 Dwell ime consrain ) i, ) ϕ, ) B κ a d i κ. 2) When any rain T arrives a a plaform i N, i dwells here for a cerain ime inerval denoed by [δ i, δ i] so ha he passengers can ge off and ge on he rain prior o is deparure from plaform. The dwell ime consrain can be wrien as follows: ) T i N δ i d i a i δ i. 3) Every rain T arrives a he firs plaform N 1) in is rain-pah eiher from he depo or by urning around from some oher line, and depars from he final plaform N N ) in order o eiher reurn o he depo or sar as a new rain on anoher line by urning around. So, he rain dwells a all plaforms in N. This is he reason why in Equaion 3) he plaform index i is varied over all elemens of he se N. 3.3 Connecion consrain In many cases, a single rain connecion migh no exis beween he origin and he desired desinaion of a passenger. To circumven his, connecing rains are ofen used a inerchange saions. Le χ N N be he se of all plaform pairs siuaed a he same inerchange saions, where passengers ransfer beween rains. Le C be he se of connecing rain pairs for a plaform pair i, ) χ. For a rain pair, ) C, rain is arriving a plaform i and rain T is deparing from plaform. A connecion ime window denoed by [χ, χ ] is mainained beween arrival of and subsequen deparure of, so ha passengers can ge off from he firs rain and ge on he laer. Le i, ) χ. Then he connecion consrain can be wrien as: ) i, ) χ, ) C χ d a i χ. 4) 5

6 3.4 Headway consrain In any railway nework, a minimum amoun of ime beween he deparures and arrivals of consecuive rains on he same rack is mainained. This ime is called headway ime. For mainaining he qualiy of passenger service, many urban railway sysem keeps an upper bound beween he arrivals and deparures of successive rains on he same rack, so ha passengers do no have o wai oo long before he nex rain comes. Le i, ) A be he rack beween wo plaforms i and, and H be he se of rain-pairs who move along ha rack successively in order of heir deparures. Consider, ) H, and le [h i, h i ] and [h ] be he ime windows ha have o be mainained beween he deparures and, h arrivals of he rains and from and o he plaforms i and respecively. So, he headway consrain can be wrien as: ) i, ) A, ) H h i d i d i h i h a a h. 5) Similarly, headway imes have o be mainained beween wo consecuive rains going hrough a crossing over. Consider any crossing over i, ) ϕ and wo such rains, which leave he erminal plaform of a rain-line i labelled as 1 and 2, raverse he crossing over i, ), and arrive a plaform of some oher rain-line labelled as 1 and 2. The se of all such rain quares 1, 1 ), 2, 2 )) is represened by H. Le [h 1 2 i, h 1 2 i ] be he headway ime window beween he deparures of rains 1 and 2 from plaform i and [h 1 2, h 1 2 ] be he headway ime window beween he arrivals of he rains 1 and 2 o he plaforms. The associaed headway consrains can be wrien as: i, ) ϕ 1, 1), 2, 2)) H ) h 1 2 i d 2 i d 1 i h 1 2 i h 1 2 a 2 a 1 h ) 3.5 Toal ravel ime consrain The rain-pah of a rain is he direced pah conaining all plaforms and racks visied by i in chronological order. To mainain he qualiy of service in he railway nework, for every rain T, he oal ravel ime o raverse is rain-pah has o say wihin a ime window [τ P, τ P ]. We can wrie his consrain as follows: T τ P a N N ) d N 1) τ ) P, 7) where N 1) and N N ) are he firs and las plaform in he rain-pah of. 3.6 Domain of he even imes Wihou any loss of generaliy, we se he ime of he firs even of he railway service period, which corresponds o he deparure of he firs rain of he day from some plaform, o sar a zero second. By seing all rip imes and dwell imes o heir maximum possible values 6

7 we can obain an upper bound for he final even of he railway service period, which is he arrival of he las rain of he day a some plaform, denoed by m Z ++. So he domain of he decision variables can be expressed by he following equaion: T i N 0 a i m, 0 d i m). 8) In vecor noaion he decision variables are denoed by a = a i ) i N ) T and d = d i) i N ) T. 4 Preliminary opimizaion model In his secion we formulae he preliminary opimizaion model ha minimizes he oal energy consumed by all rains in he railway nework. The resulan imeable is criical for he final opimizaion model, which besides minimizing energy, will also maximize he uilizaion of regeneraive energy produced by braking rains. The organizaion of his secion is as follows. Firs, in order o keep he proofs less cluered, we inroduce an equivalen consrain graph noaion. Second, we formulae and usify he preliminary opimizaion problem. Third, we show ha he nonlinear obecive of he iniial opimizaion model can be approximaed as a linear one by applying leas-squares. This resuls in a linear opimizaion problem, which has he ineresing propery ha is opimal soluion is aained by an inegral vecor. 4.1 Consrain graph noaion Each of he consrains described by Equaions 1)-7) is associaed wih wo even imes eiher arrival or deparure ime of rains a saions), where one of hem precedes anoher by a ime difference dicaed by he ime window of ha consrain. This observaion helps us o conver our iniial noaion ino an equivalen consrain graph noaion. All even imes in he original noaion are reaed as nodes in he consrain graph, he se of hose nodes is denoed by N and he value associaed wih a node i N is denoed by x i, which represens he arrival or deparure ime of some rain from a plaform. Now consider any wo nodes in he consrain graph; if here exiss a consrain beween he wo in he original noaion, hen in he consrain graph we creae a direced arc beween hem, he sar node being he firs even and he end node being he laer one. The se of arcs hus creaed in he consrain graph is denoed by Ā. Noe ha here canno be more han one arc beween wo nodes in he consrain graph. Wih each arc i, ) Ā we associae a ime window [l, u ] wih heir values deermined from he Equaions 1)-7). So, each arc i, ) Ā corresponds o a consrain of he form l x x i u. The se of all arcs associaed wih rip ime consrains is expressed by Ārip Ā. 4.2 Formulaion of he preliminary opimizaion model A rain consumes mos of is required elecrical energy during he acceleraion phase of making a rip from an origin plaform o a desinaion plaform. Trip ime consrains 7

8 play he mos imporan role in energy consumpion and regeneraive energy producion of rains. Once he rip ime for a rip is fixed, an energy opimal speed profile can be calculaed efficienly by exising sofware [15, page 3]. The elecrical power consumpion and regeneraion of a rain on a rack is deermined by is speed profile, so he opimal speed profile also gives he power versus ime graph power graph in shor) for ha rip. When a rain is driven according o he opimal driving sraegy, he elecrical energy consumed by he rain for ha rip is non-increasing in he rip ime [16]. Even when a rain is manually driven, he average energy consumpion of he rain over such manual driving sraegies is found empirically o be non-increasing in he rip ime[17]. However, in he oal railway service period here are many acive rains, whose movemens are coupled by he associaed consrains. So, finding he energy-minimal rip ime for a single rip in an isolaed manner can resul in a infeasible imeable. Consider an arc i, ) Ārip in he consrain graph, associaed wih some rip ime consrain. Le us denoe he energy consumpion funcion for ha rip f : R ++ R ++ which is non-increasing in is argumen x x i ). The preliminary opimizaion problem wih he obecive o minimize he oal energy consumpion of he rains can be wrien as: minimize f x i,) Ārip x i ) subec o i, ) Ā l x x i u, 9) i N 0 x i m, where he decision vecor is x i ) i N R N. The consrain se is affine, bu he exac analyical form of every componen of he obecive funcion for differen rips on differen racks is no known. However, in any railway nework, he amoun by which he rip ime is allowed o vary in Equaions 1) and 2) is on he order of seconds. Thus τ τ ) and κ κ ) are on he order of seconds. In such a case, we can approximae he energy funcion as an affine funcion. A reasonable approach is fiing a sraigh line hrough measured energy versus rip ime daa by leas-squares. A measuremen of he qualiy of such fiings is given by he coefficien of deerminaion, which can vary beween 0 o 1, wih 0 being he wors and 1 being he bes [18, page 518]. In our numerical sudies he mean coefficien of deerminaion of he energy fiings over all he differen rips of all he rains is found o be wih a sandard deviaion of 0.05, which usifies our approach. For energy oupu observaions f 1),..., f p) wih respec o rip imes x 1) x 1) i ),..., x p) x p) i ) respecively wihin he allowed rip ime bounds, we seek an affine funcion c x x i ) + b = x x i, 1) T c, b ) where we wan o deermine c, he slope of he line and b, he inercep wih he verical axis. The affine funcion approximaes he measured energy in he leas-squares sense as follows: c, b ) = argmin c, b ) = argmin c, b ) p k=1 c x k) x 1) x p) x k) i x 1) i, 1) T. x p) i, 1) T 8 ) + b ) 2 f k) [ c b ] f. 1) f p) )

9 The problem above is an unconsrained opimizaion problem wih convex obecive, as i is an affine mapping applied o a norm squared, and i is differeniable. So i can be solved by aking he gradien wih respec o c, b ), seing he resul equal o zero vecor and hen solving for c, b ). This yields he following closed form soluion: [ c b ] = x 1) i, 1) T x 1) x p). x p) i, 1) T T x 1) x 1) i, 1) T. x p) i, 1) T x p) 1 x 1) x 1) i, 1) T x p). x p) i, 1) T T f 1). f p) 11) Using Equaion 11), we can approximae he nonlinear obecive of he opimizaion problem 9) as an affine one: i,) Ā rip c x i x ) + b. We can also discard he b s from he obecive, as i has no impac on he minimizer. Thus we arrive a he following linear opimizaion problem o minimize he oal energy consumpion of he rains: minimize x i,) Ārip x i ) subec o i, ) Ā l x x i u, 12) i N 0 x i m. An imporan propery of his opimizaion model is ha he polyhedron associaed wih opimizaion problem has only ineger verices, so he opimal value is aained by an inegral vecor. A necessary and sufficien condiion of inegraliy of he verices of a polyhedron is given by he following heorem [19, page 269, Theorem 19.3], which we will use o prove he subsequen proposiion. Theorem 1. Le A be a marix wih enries 0, +1, or 1. For all inegral vecors a, b, c, d he polyhedron {x R n c x d, a Ax b} has only inegral verices if and only if for each nonempy collecion of columns of A, denoed by C, here exis wo subses, C 1 and C 2 such ha C 1 C 2 = C, C 1 C 2 =, and he sum of he columns in C 1 minus he sum of he columns in C 2 is a vecor wih enries 0, 1 and 1. Proposiion 1. The opimizaion problem 12) has an inegral opimal soluion. Proof. We wrie he problem 12) in vecor form. We consruc a cos vecor c, such ha a componen of ha vecor is c if i is associaed wih a rip ime consrain in he original noaion, and zero oherwise. Consruc inegral vecors l = l ) i,) Ā, u = u ) i,) Ā and marix A { 1, 0, 1} Ā N such ha he k, i)h enry of he marix A, denoed by a ki, is associaed wih he kh hyperarc and ih node of he consrain graph as follows: 1 if node i is he end node of hyperarc k, a ki = 1 if node i is he sar node of hyperarc k, 0 oherwise. So, he vecor form of he opimizaion problem 12) is: minimize c T x subec o l Ax u, 0 x m )

10 Consider any nonempy collecion of columns of A denoed by C. Take C 1 = C and C 2 =. Then he sum of he columns in C 1 minus he sum of he columns in C 2 will be a vecor wih enries 0, 1 and 1, because in A here canno exis more han one row corresponding o an arc beween wo nodes of he consrain graph and each such row has exacly wo nonzero enries, a +1 and a 1. So, by Theorem 1 he polyhedron {x R N : l Ax u, 0 x m1} has only inegral verices and opimizing he linear obecive in problem 13) over his polyhedron will resul in an inegral soluion. Afer solving he linear programming problem 12), we obain an inegral imeable, which we will call he energy minimizing imeable EMT). We denoe he opimal decision vecor of his imeable by x in he consrain graph noaion and ā i, d i ) i N ) T in he original noaion. 5 Final opimizaion model In his secion we modify he rip ime consrains such ha he oal energy consumpion of he final imeable is kep a he same minimum as he EMT. Then, we describe our opimizaion sraegy aimed o maximize he uilizaion of regeneraive energy of braking rains, and we presen he final opimizaion model. 5.1 Keeping he oal energy consumpion a minimum In any feasible imeable, if he rip imes are kep o be he same as he ones obained from he EMT, hen he energy opimal speed profiles for all rains will be he same. As a resul, he energy consumpion associaed wih ha imeable will remain a he same minimum as found in he EMT. So, in he final opimizaion problem, insead of using he rip ime consrain, for every rip we fix he rip ime o he value in he EMT, i.e., and T i, ) A a d i = ā d i), 14) i, ) ϕ, ) B a d i = ā d i ). 15) For all oher consrains, bounds are allowed o vary as described by Equaions 3)-7). As a consequence of fixing all rip imes, he power graph of every rip made by any rain becomes known o us, since i depends on he corresponding opimal speed profile calculaed by exising sofware [15, page 3]. 5.2 Maximizing he uilizaion of regeneraive energy of braking rains In his subsecion we describe our sraegy o maximize he uilizaion of he regeneraive energy produced by he braking rains. Sraegies based on ransfer of regeneraive braking 10

11 i Braking Dwelling i Acceleraing Power a i a i i ) d i + i ) d i Maximum power consumpion Maximum power consumpion e a i a i i ) d i d i + i ) Regeneraive alignmen poin Maximum power regeneraion e Consumpive alignmen poin Time Maximum power regeneraion Figure 1: Applying 1 e heurisic o power graphs energy back o he elecrical grid requires specialized echnology such as reversible elecrical subsaions [20, page 30]. A sraegy based on soring is no feasible wih presen echnology, because sorage opions such as super-capaciors, fly-wheels, ec., have drasic discharge raes besides being oo expensive [21, page 66], [22, page 92]. A beer sraegy ha can be used wih exising echnology [23] is o ransfer he regeneraive energy of a braking rain o a nearby and simulaneously acceleraing rain, if boh of hem operae under he same elecrical subsaion. We call such pairs of rains suiable rain pairs. So our obecive is o maximize he oal overlapped area beween he graphs of power consumpion and regeneraion of all suiable rain pairs. To model his mahemaically, we are faced wih he following asks: i) define suiable rain pairs, ii) provide a racable descripion of he overlapped area beween power graphs of such a pair. We describe hem as follows Defining suiable rain pairs We consider plaform pairs who are opposie o each oher and are powered by he same elecrical subsaions. Thus, he ransmission loss in ransferring elecrical energy beween hem is negligible. The se conaining all such plaform pairs is denoed by Ω. Consider any such plaform pair i, ) Ω, and le T i T be he se of all rains which arrive a, dwell 11

12 Line 2 PES2GRW2LXM2 LJB2 SXZ2 ZJD2 YHR2 CSR2YSS2JYR2 LZV2 LHR2 PJT2 JYS2LHS2 SFM2 GRW1 LXM1LJB1 SXZ1 ZJD1 YHR1 CSR1 YSS1JYR1 LZV1LHR1 PJT1 JYS1 LHS1 Line 1 Figure 2: Railway nework considered for numerical sudy and hen depar from plaform i. Suppose, T i. Now, we are ineresed in finding anoher rain on plaform, i.e., T, which along wih would form a suiable pair for he ransfer of regeneraive braking energy. To achieve his, we use he EMT. Among all rains going hrough plaform, he one which is emporally closes o in he energy-minimizing imeable is be he bes candidae o form a pair wih. The emporal proximiy can be of wo ypes wih respec o, which resuls in he following definiions. Definiion 1. Consider any i, ) Ω. For every rain T i, he rain T is called he emporally closes rain o he righ of if { ā = argmin i + d } i ā + d, 16) 2 2 {x T :0 āx + d x 2 ā i + d i 2 r} where r is an empirical parameer deermined by he imeable designer and is much smaller han he ime horizon of he enire imeable. Definiion 2. Consider any i, ) Ω. For every rain T i, he rain T is called he emporally closes rain o he lef of if { ā = argmin i + d } i ā + d. 17) 2 2 {x T :0< ā i + d i 2 āx + d x 2 r} Definiion 3. Consider any i, ) Ω. For every rain T i, he rain T is called he emporally closes rain o if { ā i = argmin + d } i ā + d. 18) 2 2 {, } If boh and are emporally equidisan from, we pick one of hem arbirarily. Any synchronizaion process beween a suiable rain pair SPSTP) can be described by specifying he corresponding i,, and by using he definiions above. We consruc a se of all SPSTPs, which we denoe by E. Each elemen of his se is a uple of he form i,,, ). Because is unique for any in each elemen of E, we can pariion E ino wo ses denoed by E and E conaining elemens of he form i,,, ) and i,,, ) respecively. 12

13 5.2.2 Descripion of he overlapped area beween power graphs The power graph during acceleraing and braking is highly nonlinear in naure wih no analyic form, as shown in Figure 1. So, maximizing he exac overlapped area will lead o an inracable opimizaion problem. However, he exisence of dominan peaks wih sharp falls allows us o apply a robus lumping mehod such as 1 heurisic [24, page 33- e 34] o approximae he power graphs as recangles. The 1 heurisic is applied as follows e see Figure 1). The heigh of he recangle is he maximum power, and he widh is he inerval wih exreme poins corresponding o power dropped a 1/e of he maximum. For he sharp drop from he peak, such recangles are very robus approximaions o he original power graph conaining he mos concenraed par of he energy, e.g., if he drop were exponenial, hen he energy conained by he recangle would have been exacly equal o ha of he original curve [24, page 33-34]. Afer convering boh he power graphs o recangles, maximizing he overlapped area under hose recangles is equivalen o aligning he midpoin of he widh of he recangles; we call such a midpoin regeneraive or consumpive alignmen poin. These alignmen poins ac as virual peaks of he approximaed power graphs. As shown in Figure 1, for a rain in is braking phase prior o is arrival a plaform i, he relaive disance of a i from he regeneraive alignmen poin is denoed by i, while during acceleraion he relaive disance of he consumpive alignmen poin from d i is denoed by. Noe ha boh relaive disances are known parameers for he curren opimizaion problem. 5.3 Final opimizaion model Consider an elemen i,,, ) E. To ensure he ransfer of maximum possible regeneraive energy from he braking rain o he acceleraing rain, we aim o align boh heir alignmen poins such ha d i + i = a, or keep hem as close as possible oherwise. Similarly, for any i,,, ) E, our obecive is d + = a i i, or as close as possible. Le a decision vecor y be defined as y = d i + i a + ) i,,, ), d E + a i + ) i i,,, ) E ). 19) Then our goal comprises of wo pars: 1) maximize he number of zero componens of y which corresponds o minimizing cardy), and 2) keep he nonzero componens as close o zero as possible which corresponds o minimizing he l 1 norm of y, y 1. Combining hese wo we can wrie he exac opimizaion problem as follows: minimize cardy) + γ y 1 subec o Equaions3) 7), 14), 15), 19), T i N 0 a i m, 0 d i m), where γ is a posiive weigh, and decision variables are a, d and y. The obecive funcion is nonconvex as shown nex. Take he convex combinaion of he vecors 2e 1 /γ and 0 wih 13 20)

14 convex coefficiens 1/2. Then, ) e1 card + γ e 1 γ γ = 2 > card ) 2e1 + γ γ 2e 1 γ ) card 0) + γ 0 1 ) = 1.5, and hus violaes definiion of a convex funcion. As a resul, problem 20) is a nonconvex problem. Noe ha if we remove he cardinaliy par from he obecive, hen i reduces o a convex opimizaion problem because he consrains are affine and he obecive is he l 1 norm of an affine ransformaion of he decision variables [25, pages 72, 79, ]. Such problems are ofen called convex-cardinaliy problem and are of N P-hard compuaional complexiy in general [26]. An effecive ye racable numerical scheme o achieve a lowcardinaliy soluion in a convex-cardinaliy problem is he l 1 norm heurisic, where cardy) is replaced by y 1, hus convering problem 20) ino a convex opimizaion problem. This is described by problem 21) below. The l 1 norm heurisic is suppored by exensive numerical evidence wih successful applicaions o many fields, e.g., robus esimaion in saisics, suppor vecor machine in machine learning, oal variaion reconsrucion in signal processing, compressed sensing ec.. In he nex secion we show ha in our problem oo, he l 1 norm heurisic produces excellen resuls. Inuiively, he l 1 norm heurisic works well, because i encourages sparsiy in is argumens by incenivizing exac alignmen beween regeneraive alignmen poins wih he associaed consumpive ones [25, pages ]. We provide a heoreical usificaion for he use of l 1 norm in our case as follows. Proposiion 2. The convex opimizaion problem described by minimize y 1 subec o Equaions3) 7), 14), 15), 19), T i N 0 a i m, 0 d i m), 21) is he bes convex approximaion of he nonconvex problem 20) from below. Proof. Boh problems 21) and 20) have he same consrain se, so we need o focus on he obecive only. The bes convex approximaion of a nonconvex funcion f : C R where C is any se) from below is given by is convex envelope env f on C. The funcion env f is he larges convex funcion ha is an under esimaor of f on C, i.e., env f = sup{ f : C R f is convex and f f}, where sup sands for he supremum, i.e., he leas upper bound of he se. The definiion implies, epi env f = conv epi f. From Equaion 19) we see ha y is an affine ransformaion of a and d, and from he las consrains of problem 20) we see ha boh a and d are upper bounded by m, i.e., a m and d m. So here exiss a posiive number P such ha y P. As he domain of y is bounded in an l ball wih radius P, env cardy) = 1 P y 1 [27, page 321]. So, he bes convex approximaion of he obecive from below is 1 P y 1 + γ y 1 = 1 P + γ) y 1. 14

15 As he coefficien 1 + γ) is a consan for a paricular opimizaion problem, i can be P omied, and hus we arrive a he claim. Using he epigraph approach [25, pages ], we can ransform he convex problem 21) ino a linear program as follows. For each i,,, ) E and each i,,, ) E, we inroduce new decision variables θ and θ respecively, such ha θ d i + i a + and θ d + a i + i. Then, he convex opimizaion problem can be convered ino he following linear problem: minimize subec o i,,, ) E i,,, ) E i,,, ) E i,,, ) E θ θ θ θ θ + i,,, ) E θ d i+ i a d i i +a + ), ), d + a i + i), d +a i i), i,,, ) E Equaions3) 7), 14), 15), T i N 0 a i m, 0 d i m), 22) where he decision variables are a i, d i, θ and θ. 6 Numerical sudy In his secion we apply our model o differen problem insances spanning full service period of one day o service PES2-SFM2 of line 8 of Shanghai Mero nework. The numerical sudy was execued on a Inel Core i CPU wih 8 GB RAM running Windows 8.1 Pro operaing sysem. For modelling he problem, we have used JuMP - an open source algebraic modelling language embedded in programming language Julia [28]. Wihin our JuMP code we have called academic version of Gurobi Opimizer 6.0 as he solver. We have implemened an inerior poin algorihm because of he underlying sparsiy in he daa srucure. As menioned before, a measure of he qualiy of affine fiings using leas-squares approach is given by he coefficien of deerminaion, which can vary beween 0 o 1, wih 0 being he wors and 1 being he bes [18, page 518]. In our numerical sudy, he average coefficien of deerminaion of he affine fiings for energy versus rip imes over all differen rips and all rains is found o be wih a sandard deviaion of 0.05, which usifies our approach. The service PES2-SFM2 of line 8 of he Shanghai Mero nework is shown in Figure 2. There are wo lines in his nework: Line 1 and Line 2. There are foureen saions in he nework denoed by all capialized words in he figure. Each saion has wo plaforms each on differen rain lines, e.g., LXM is saion ha has wo opposie plaforms: LXM1 and LXM2 on Line 1 and Line 2 respecively. The plaforms are denoed by recangles. The 15

16 Number of rains Number of consrains Sage 1 Number of variables Sage 1 Table 1: Resuls of he numerical sudy Sage 1 CPU ime s) Number of consrains Sage 2 Number of Variable Sage 2 Sage 2 CPU Time s) Iniial effecive energy consumpion kwh) Final effecive energy consumpion kwh) Reducion in effecive energy consumpion % % % % % % % % % % % plaforms indicaed by PES2 and SFM2 are he urn-around poins on Line 2, wih he crossing-overs being PES2-GRW1 and LHS1-SFM2. The duraion of he imeables is eigheen hours which is he full service period of he railway nework. We have considered eleven differen insances wih varying average headway imes and number of rains. The number of rains increases as he average headway ime decreases. The resuls of he numerical sudy are shown in Table 1. We can see ha, in all of he cases our model has found he opimal imeables very quickly, he larges runime being 12.58s. To he bes of our knowledge, his model is he only one o calculae energy-efficien railway imeable spanning an enire day, he nex larges being 6 hours only [8] wih a much larger compuaion ime for smaller sized problems. Afer we ge he final imeable, we calculae he oal effecive energy consumpion by all rains involved in SPSTPs and compare i wih he original imeables. The effecive energy consumpion of a rain during a rip is defined as he difference beween he oal energy required o make a rip and he amoun of energy ha is being supplied by a braking rain during synchronizaion process. So, he effecive energy consumpion is he energy ha will be consumed from he elecrical subsaions. The original imeables, which we compare he final imeables wih, are provided by Thales Canada Inc. The energy calculaion is done using SPSIM, which is a proprieary sofware owned by Thales Canada Inc [15], and Cubaure, which is an open-source Julia package wrien by Seven G. Johnson [29]. SPSIM calculaes he power versus ime graphs of all he acive rains for he original and opimal imeables. Cubaure is used o calculae he effecive area under he power versus ime graphs o deermine 1) he oal energy required by he rains during he rips, 2) he oal ransferred regeneraive energy during he SPSTPs, and 3) he effecive energy consumpion as he difference of he firs wo quaniies. The effecive energy consumpion of he opimal imeables in comparison wih he original ones 16

17 is reduced quie significanly - even in he wors case, he reducion in effecive energy consumpion is 19.27%, wih he bes case corresponding o 21.61%. 7 Conclusion In his paper we have proposed a novel wo-sage linear opimizaion problem o calculae an energy-efficien imeable in modern railway neworks. The obecive is o minimize he oal elecrical energy consumpion of all rains and o maximize he uilizaion of regeneraive energy produced by braking rains. In conras o oher exising models, his model is compuaionally he mos racable one, and can be applied o any railway nework using exising echnology. We have applied our opimizaion model o eleven differen insances of service PES2-SFM2 of line 8 of Shanghai Mero nework. All insances span he full service period of one day 18 hours) wih housands of acive rains. For all insances our model has found opimal imeables in less han 13s wih significan reducions in he effecive energy consumpion. Acknowledgemens This work was suppored by NSERC-CRD and Thales, Inc CRDPJ ). The auhors acknowledge helpful discussions wih Professor J. Chrisopher Beck, Deparmen of Mechanical & Indusrial Engineering, Universiy of Torono. References [1] R. Pail, J. C. Kelly, Z. Filipi, and H. Fahy, A framework for he inegraed opimizaion of charging and power managemen in plug-in hybrid elecric vehicles, in American Conrol Conference ACC), 2012, pp [2] T. Nuesch, T. O, S. Ebbesen, and L. Guzzella, Cos and fuel-opimal selecion of hev opologies using paricle swarm opimizaion and dynamic programming, in American Conrol Conference ACC), June 2012, pp [3] R. Mura, V. Ukin, and S. Onori, Recasing he hev energy managemen problem ino an infinie-ime opimizaion problem including sabiliy, in IEEE 52nd Annual Conference on Decision and Conrol CDC), 2013, pp [4] S. Bashash, S. J. Moura, J. C. Forman, and H. K. Fahy, Plug-in hybrid elecric vehicle charge paern opimizaion for energy cos and baery longeviy, Journal of Power Sources, vol. 196, no. 1, pp , [5] A. Y. Saber and G. K. Venayagamoorhy, Inelligen uni commimen wih vehicleo-grida cos-emission opimizaion, Journal of Power Sources, vol. 195, no. 3, pp ,

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