ANALYTICAL PROBABILISTIC MODELING OF URBAN DRAINAGE SYSTEMS

Transcription

1 ANALYTICAL ROBABILITIC MODELING OF URBAN DRAINAGE YTEM B.J. Aams* an F. apa** * Dept. of Civil Engineeing, Univesity of Toonto, Toonto, Ontaio, Canaa M5 1A4 Tel (1) , Fax (1) , aams@civ.utoonto.ca ** Valo Engineeing Inc., 216 Chislea Roa, uite 501, Woobige, Ontaio, Canaa L4L 85 Tel (1) , Fax (1) , fpapa@valo-engineing.com KEYWORD: tomwate Management lanning, Runoff Contol, obabilistic Moels UMMARY The avese envionmental impacts of uncontolle uban ainage systems ae well ocumente. Infome envionmental ecision making equies knowlege of impacts esulting fom combine sewe oveflows (COs) an stomwate ischages, the ability of contol technologies to mitigate these impacts, an thei associate costs. Consequently, the moelling of uban ainage systems is of impotance to this pocess. While seveal moeling appoaches have been evelope to aess these issues, each of which have thei avantages an isavantages, they typically equie extensive input an significant computational effot. The moels pesente heein ae highly computationally efficient an ae intene to enhance planning-level analyses an the evelopment of altenative solutions to uban ainage poblems. This moeling appoach employs statistics eive fom long-tem ainfall ecos to efine pobability ensity functions (DFs) fo ainfall chaacteistics. Though elatively simple mathematical epesentations of the tansfomation of ainfall to unoff an the geneation of pollutant washoff fom a catchment, the DFs of unoff chaacteistics ae eive fom the DFs of ainfall chaacteistics. Contol elements, such as stoage facilities, have been incopoate into the moel evelopments an the DFs of system pefomance vaiables can similaly be eive. Examples of system pefomance statistics of impotance in analyzing options fo CO contol inclue the aveage annual numbe of oveflows an the aveage annual volume an pollutant mass of oveflows as well as the volumes an pollutant masses of exteme events. If teatment is employe, statistics such as the level of unoff an pollution contol pefomance ae impotant an can be easily calculate. The eal powe offee by this moelling appoach is the ability to pouce meaningful esults quickly an easily. This featue enhances the moelling expeience by facilitating sensitivity analyses an, though the incopoation of cost functions, system optimization. Analytical obabilistic Moeling of Uban Dainage ystems age 1

2 INTRODUCTION Uban infastuctue, incluing stomwate ainage systems, emans massive commitments of capital fo constuction, opeation an maintenance. In oe to aequately maintain a given level of sevice, such as floo potection an/o wate quality contol, while minimizing the esouces equie to o so, moels fo the pefomance analysis of stomwate ainage systems ae essential. Moels ae, howeve, only a epesentation of the physical eality an simplifying assumptions ae invaiably equie to evelop tactable moels of ainage system pefomance. Geneally, the geate the egee of accuacy of the moel, the geate is its level of complexity an, hence, computational buen. A balance, theefoe, must be stuck between the accuacy an the simplicity of the moel; this balance is epenent on the pupose to be seve by the moel. That is, the selection of moels an moelling complexity shoul eflect the type of analysis equie. The analytical pobabilistic moels popose heein ae intene fo sceening an planning level analyses to povie immeiate insight into the magnitue of stomwate poblems, to illustate unoff contol option tae-offs, an geneally to assess system pefomance. Event-base esign stom methos, especially those using intensity-uation-fequency elations an synthetically geneate ainfall pattens, have been taitionally employe in the esign an analysis of stomwate conveyance systems (see Figue 1). uch methos, howeve, cannot eveal infomation egaing the aveage, long-tem pefomance of ainage systems, especially those systems employing stoage. This unestaning can only be gleane fom a continuous analysis of system pefomance. Futhemoe, unoff quality an eosion contol pefomance can only be aequately assesse by continuous analysis as these impacts ae lagely influence by smalle, moe fequent ainfall events an, hence, ae moe stongly felt in the long tem. The most commonly employe appoach fo long tem ainage system pefomance analysis is continuous simulation moeling which tansfoms long tem hyetogaphs, eive fom ecos of obseve ainfall ata, into unoff hyogaphs an outes these hyogaphs though conveyance systems, stoage facilities an othe contol evices. The esult is a time seies of ainage system esponses, incluing flow ates an stoage level fluctuations, which ae then statistically analyze to povie infomation about the pefomance of the ainage system (see Figue 1). Continuous simulation moels can become vey complex an may equie elatively lage commitments of esouces. As an altenative to sceening an planning level analysis of ainage systems by continuous simulation, analytical pobabilistic moels have been popose an implemente. These moels ae evelope by pepocessing meteoological ecos to ceate pobability ensity functions (DFs) of meteoological chaacteistics (e.g. ainfall event volumes, uations, inteevent times, etc.) an then tansfoming these DFs of system inputs to DFs of system outputs though the hyological/hyaulic tansfomation functions of the catchment an its contol evices. ince the DFs of meteoological inputs ae eive fom the statistical analysis of long tem ainfall ecos, the mathematically eive DFs of system outputs eflect the long tem pefomance of the ainage system une analysis. Analytical obabilistic Moeling of Uban Dainage ystems age 2

3 Figue 1. Appoaches to Dainage ystem Analysis A significant outcome of this pocess is the eivation of these DFs, an thei associate statistics of unoff contol system pefomance, in mathematically close fom. These algebaic equations explicitly elate the pobabilities of pefomance vaiables to Analytical obabilistic Moeling of Uban Dainage ystems age 3

4 meteoologic, hyologic an esign paametes thus eneing the exploation of the complete omains of paamete values an esign altenatives computationally simple. In oe to obtain this mathematical flexibility, analytical moels utilize moe simplifying assumptions than compaable simulation moels. Theefoe, when the accuacy of an analysis can be elaxe, pemitting the use of elatively simple an, hence, inexpensive analysis techniques, these analytical pobabilistic moels povie a poweful an compehensive tool not othewise available at the sceening an planning levels. ROBLEM TYE AND MITIGATION MEAURE As a esult of the ubanization of lan, unoff volumes an flow ates geneally incease an, if left uncontolle, ae liable to cause flooing an eosion poblems ownsteam. tomwate unoff stoage facilities, such as etention pons an unegoun tanks, ae an effective means of attenuating peak unoff ates such that they ae moe compatible with the capacity of the ownstean ainage system. Eosion impacts can be mitigate by alteing the istibution of flow uations though the use of stoage facilities an outlet flow contols. ewes susceptible to stomwate flows ae liable to expeience suchage conitions an, whee such sewes have esiential house connections, may thus esult in basement flooing. In the case of combine sewe systems, sewe sepaation an the intouction of stoage esevois ae common mitigative measues. Wate quality egaation esulting fom uban ainage, in the fom of combine sewe oveflows an stom sewe ischages, is a topic which has eceive much ecent attention. Combine sewe oveflow volumes may be euce by employing stoage esevois which elease flows at ates moe compatible with the combine sewe system an ae ultimately teate at wastewate teatment plants. The etention povie in stoage facilities can be futhe capitalize upon by poviing teatment to stomwate unoff an/o combine sewage. Typical emoval mechanisms pimaily inclue seimentation, bacteial ecay an biological uptake. ANALYI NEED Thee is a wie ange of altenatives to mitigate the poblems associate with uban ainage systems. Diffeent altenatives ae moe o less effective at mitigating iffeent ainage system poblems to iffeent egees. The challenge to the enginee/analyst is to make ecisions on which altenative (o combination of altenatives) to eploy, at what scale of implementation, an une what cicumstances. This ecision equies infomation of the effectiveness of the altenatives, an thei costs, whee effectiveness is usually etemine by pefomance moelling of the altenatives. efomance moels must be capable of peicting behaviou une exteme events as well as fo annual aveage conitions; hence, the nee fo continuous analysis of uban ainage systems. Analytical obabilistic Moeling of Uban Dainage ystems age 4

5 CATCHMENT MODEL The hyologic moels evelope heein ae intene fo sceening-level analysis. It is impotant to unestan the simplifying assumptions on which they ae base. The fist assumption egas the natue of the ainage system an the mechanics of wate movement, which ae illustate in Figue 2. Figue 2. chematic moel of uban ainage systems. Beginning with the meteoological input to the catchment, it is assume to be epesente by the exponential DFs of the ainfall chaacteistics. The catchment then tansfoms the ainfall volume, v (mm), to unoff volume, v (mm), accoing to the following elationship, which is the hyologic pesentation employe by the TORM simulation moel (U.. Amy Cops of Enginees, 1974) hence, the moels eive fom the elationship ae efee to as the ATORM moels (fo Analytical TORM): v 0 ( v ) ; v ; v > whee ainfall must fill the volume of epession stoage, (mm), befoe unoff occus. Fo ainfall volumes above, the unoff volume is given by a pouct of a imensionless unoff coefficient,, an the excess of ainfall ove epession stoage, (v ). The volume (1 - )(v - ) may be viewe as a unifom infiltation loss occuing afte the initial epession stoage loss an lasting until the en of the event. The unoff coefficient,, is a spatially an tempoally aveage constant which is selecte on the basis of lan use, soil type, an topogaphy. It is assume that the uation of the unoff event, t, is pactically equal to the uation of the ainfall event, t. It is also note that the full epession stoage,, is assume to be available at the beginning of evey ainfall event since the volume of epession stoage on uban catchments is typically small. Once the unoff is geneate as a squae-wave hyogaph, it is tanslate to the esevoi instantaneously, without change in shape. The ationale fo this assumption is that fo unoff contol systems incopoating stoage, the system pefomance is influence moe by the (1) Analytical obabilistic Moeling of Uban Dainage ystems age 5

6 unoff volume an uation than by the exact shape of the unoff hyogaph. It is note that the moels pesente heein ae intene fo the analysis of uban ainage systems with stoage. The esign of ainage conveyances (e.g., pipes, channels), howeve, epens on the peak ate of flow an ae thus sensitive to the shape of the unoff hyogaph. Fo such poblems, altenative analytical moels of the AWMM Type (e.g., Guo an Aams, 1998a, 1998b; 1999a, 1999b, 1999c) o altenative simulation moels woul be moe appopiate. The maximum stoage volume of the esevoi, A (mm), is expesse as the equivalent epth acoss the catchment aea. The esevoi is aine by a contolle outflow at a constant ate, (mm/h), also expesse as equivalent epth acoss the catchment aea. When the esevoi is full an the inflow occus at a geate ate than the outflow, the excess volume, p (mm), is spille. uch spills may be visualize as occuing upsteam of the stoage facility (though a flow splitte/egulato) o as being oute though the stoage facility. ROBABILITY DENITY FUNCTION OF RAINFALL EVENT CHARACTERITIC Chaacteistics of ainfall events of inteest in ainage systems analysis ae: event volume (v), event uation (t), aveage intensity of event (i), an the inteevent time (b) which is the y peio between successive ainfall events. In oe to evelop the pobability ensity functions (DFs) of these chaacteistics, iniviual ainfall events ae ientifie by the citeion of the minimum inteevent time efinition (IETD). Events with inteevent times geate than o equal to this minimum ae classifie as istinct events. With this citeion, a long tem ainfall eco (typically ecaes in length) is pase into events. The samples of values of each chaacteistic ae then subjecte to a statistical analysis evealing thei moments an coefficients. ample histogams an/o cumulative istibution functions (CDFs) ae plotte an theoetical istibutions ae fitte. In this wok, exponential istibutions ae fitte to the sample ata as summaize in Table 1. In aition to the numeical values of, λ, β an ψ (the inveses of the mean ainfall event volume, uation, aveage intensity an inteevent time, espectively), the value of θ, the aveage annual numbe of ainfall events, is also known fom the statistical analysis of the ainfall eco. Analytical obabilistic Moeling of Uban Dainage ystems age 6

7 Table 1. DFs of ainfall chaacteistics Rainfall Chaacteistics Exponential DF Applicable Range Volume, v (mm) v f ( v) e V 1 v 0 v Duation, t (h) t f ( t) λ e λ T 1 λ t 0 t Aveage intensity, i (mm/h) t f ( i) β e β I 1 β i 0 i Inteevent time, b (h) implifie vesion f B ( b ) ψ e ψ f B (b) ψ e -ψ b ( bietd) ψ ψ 1 b-ietd 1 b IETD b 0 b DERIVED ROBABILITY DITRIBUTION THEORY To illustate the pinciple of eive pobability istibution theoy, a simple case of a function of a single anom vaiable is iscusse below. Consie a anom vaiable X with a known pobability ensity function (DF), f X (x). Also consie a monotonically inceasing function, y g(x), which tansfoms values of X to values of Y in one-to-one coesponence (i.e., only one value of Y exists fo evey value of X an vice vesa). Then Y is a anom vaiable whose DF, f Y (y), can be eive as follows (see Figue 3). If y g(x) [theefoe, x g -1 (y)] is monotonically inceasing an maps X Y one-to-one, then F Y -1 ( y) ob[ Y y] ob X g ( y) 1 [ ] F [ g ( y) ] whee F Y (y) is the cumulative istibution function (CDF) of Y. By efinition, the DF is the fist eivative of the CDF, o 1 g ( y) [ ] f ( x) 1 ( ) ( ) ( ) f Y y FY y FX g y X x (3) y y y Using Leibniz s ule fo iffeentiating an integal, Equation 3 is equivalent to f Y [ ] 1 1 ( y) g ( y) f g ( y) X X (4) y (2) Analytical obabilistic Moeling of Uban Dainage ystems age 7

8 Figue 3. Gaphical intepetation of eive pobability istibution of a monotonic function of one anom vaiable with a one-to-one mapping. (Afte Benjamin an Conell, 1970). o, since x g -1 (y) x f Y y ( y) f ( x) Restating Equation 5 yiels Y X ( y) y f ( x)x f X (5) (6) The esult above is illustate gaphically in Figue 3. If y g(x) maps X Y one-to-one but is monotonically eceasing, then Equations 4 an 5 ae moifie espectively to an f Y y [ ] 1 1 ( y) g ( y) f g ( y) X (7) x f Y y ( y) f ( x) X (8) Analytical obabilistic Moeling of Uban Dainage ystems age 8

9 EXAMLE OF DERIVED ROBABILITY DITRIBUTION RUNOFF VOLUME Given the maginal DF of ainfall volume (Table 1), an the unoff moel (Equation 1), the cumulative istibution function (CDF) of unoff volume, F V (v ), may be obtaine using eive pobability istibution theoy. Fom the CDF of unoff volume, the DF of unoff volume, f V (v ), may be obtaine by iffeentiation. Figue 4 schematically shows the tansfomation of the DF of ainfall volume to the DF of unoff volume accoing to the ainfall-unoff tansfomation function as epicte in Equation 1. Accoing to the ainfallunoff moel, some ainfall events will not cause unoff events if thei volume is less than the epession stoage. As a esult, thee is an impulse pobability that no unoff will occu which is equal to the pobability that a given ainfall event s volume oes not excee epession stoage which is epesente by the shae aea in Figue 4. This impulse pobability is given by p V ( 0) ob[ V 0] ob[ V ] f ( v) v e 1 e v 0 V v v (9) v 0 The emaine of the CDF of unoff volume exists ove the ange whee unoff occus (v > 0) which coespons to the ange whee the volume of a ainfall event is geate than the epession stoage value (v > ). That is, Figue 4. Tansfomation of DF of ainfall volume to DF of unoff volume Analytical obabilistic Moeling of Uban Dainage ystems age 9

10 F V ( v ) ob[ V v ] ob( V 0) ob ob ( V 0) f ( v) ( V 0) ove the ange whee v > 0. v 1 e V v v e ob 1 e < V v v (10) The DF of unoff volume may be obtaine as the eivative of Equation 10 as follows: v v f V v F > V v v ( ) 1 e v e ; v 0 The impulse pobability associate with v 0 is note an is given by Equation 9. (11) YTEM ERFORMANCE MODEL As with the eivation of the DF of unoff volume, eive pobability istibution theoy is use to evelop the DFs of numeous pefomance vaiables that escibe the behaviou of the catchment an the unoff quantity an quality contol system. In aition to the full DFs of the pefomance vaiables, the event means an the aveage annual means of the pefomance vaiables ae also eive. A summay of the esultant pefomance moels is given below. The mathematical eivations of these moels is given by Aams an apa (2000). It is note that the pefomance moels ae eive fo two sets of assumptions egaing esevoi contents: (1) esevoi full at the en of the last event (s i A ) an (2) esevoi empty at en of last event (s i 0). In aition, two sets of equations ae eive fo the two foms of the DF fo inteevent time as given in Table 1. The summay below inclues the pefomance moels evelope fo the simple DF of inteevent time. RECIITATION Aveage annual pecipitation volume θ p (12) RUNOFF Aveage annual volume of unoff Analytical obabilistic Moeling of Uban Dainage ystems age 10

11 Analytical obabilistic Moeling of Uban Dainage ystems age 11 R θ e (13) Aveage annual numbe of unoff events n θ e (14) Aveage annual loss volume ( ) L θ e 1 (15) Aveage annual volume of epession stoage loss ( ) D a θ e 1 (16) RUNOFF QUANTITY CONTROL obability pe ainfall event of spill with a magnitue of at least p 0 (assuming that s i A ) ( ) A p e p ψ ψ ψ λ λ 0 e G 0 (17) obability pe ainfall event of spill with a magnitue of at least p 0 (assuming that s i 0) ( ) A p p λ λ 0 e G 0 (18) obability pe ainfall event of any spill occuing (assuming that s i A ) ( ) A ψ ψ ψ λ λ e e 0 G (19)

12 obability pe ainfall event of any spill occuing (assuming that s i 0) λ ( ) A G 0 e (20) λ Aveage annual numbe of spills n s θ G ( 0) (21) Aveage annual spill volume u θ G p ( 0) ns (22) pill volume of specifie etun peio, T R (yeas) ln pt R T R [ θ G ( 0) ] (23) Aveage annual faction of unoff lost to spills R s u R (24) Aveage annual faction of unoff contolle C R 0 R u 1 1 G e (25) Isoquant of aveage annual numbe of spills, n s A n s ψ ψ ln 1 e assuming s i A (26) ψ θ λ A n ln assuming s i 0 (27) θ λ s 1 Analytical obabilistic Moeling of Uban Dainage ystems age 12

13 Isoquant of aveage annual faction of unoff contolle, C R ψ ψ A ln ( 1 CR ) 1 assuming s i A (28) ψ λ ( 1 CR ) 1 A ln assuming s i 0 (29) λ Aveage annual numbe of unoff events that o not utilize stoage β β Φ n θ e - p 1 e whee Φ (30) β Aveage annual volume of unoff pocesse without stoage o β Φ β θ e β w 1 1 e whee Φ (31) λβ β w R u θ [ e G( 0) ] RUNOFF QUALITY CONTROL Combine pollution emoval efficiency ( η ) η η η η η η η 1 (32) Aveage annual faction of pollution contol C ( R ) u η u Tη R (33) Aveage steay-state etention time of stoage facility t A s t (34) Analytical obabilistic Moeling of Uban Dainage ystems age 13

14 Oveall factional T emoval efficiency une ynamic settling conitions E i n V i A Fi 1 1 (35) nha 2 Oveall factional T emoval efficiency une quiescent settling conitions E q i V i F i ψ h ψ h Vi 1 e (36) Aveage annual faction of pollution contol fom extene etention y pon C E C R E [ 1 G ( 0) e ] (37) Aveage annual faction of pollution contol fom wet pon without outlet contol C 1 e (38) Eq Aveage annual faction of pollution contol fom wet pon with outlet contol (i.e. extene etention wet pon) whee C E q 1 e ' u θ G ( 0) ( ' ) an G (0) is calculate using effective epession stoage effective u R u E (39) (40) (41) The mathematically compact fom of the above moel expessions fo system pefomance offes exceptional ease in geneating esults an pefoming sensitivity analyses in oe to gain an oveall unestaning of system behaviou an illustating contol option tae-offs. Futhemoe, these pefomance moels of stomwate management contol altenatives may be easily combine with economic functions of system esign vaiables in an optimization Analytical obabilistic Moeling of Uban Dainage ystems age 14

15 context. The esult of this execise is the fomulation of methoologies fo the cost-effective selection of esign altenatives an scale of implementation of altenatives to meet unoff quantity an quality contol objectives of uban ainage systems. VALIDITY OF ERFORMANCE MODEL The above pefomance moels ae evelope with a numbe of simplifying assumptions fo easons of mathematical tactability. The evelopments geneally follow the pocesses of hyologic simulation moels (i.e., the TORM moel in the pesent wok) but can neve inclue the level of complexity allowe by simulation moeling. Howeve, this is consistent with the intene use of the analytical pefomance moels fo system planning, sceening level analysis, an peliminay esign. Nevetheless, it is impotant to unestan the limits of accuacy an the ange of applicability of the moels. Theefoe, extensive compaisons have been unetaken between the esults obtaine fom the analytical pefomance moels an the equivalent esults fom continuous simulation moels (e.g., eto, 1984; Kauffman, 1987; apa et al., 1997; Li an Aams, 2000). In geneal, favouable compaisons have been foun fo many climatic egions. These esults suggest that analytical pefomance moels fo unoff quantity an quality contol planning offe a poweful altenative/supplement to continuous simulation moeling. CONCLUION This pape pesents a summay of a suite of analytical pobabilistic moels cuently available fo the planning level analysis an esign of uban ainage systems. These pefomance moels aess the most funamental poblems face by the enginee/planne. Thei mathematical close fom allows fo exteme computational efficiency an ease of implementation. They povie immeiate insight into the pefomance of uban ainage systems an the effectiveness of unoff contol altenatives in mitigating typically encountee poblems such as flooing, eosion an wate quality egaation. These attibutes enhance the engineeing analysis of uban ainage systems by encouaging the compehensive analysis of wie anges of system esign altenatives which, in tun, impoves the cost effectiveness of system esign. These moels ae applie in a companion pape (apa an Aams, 2002) to a vaiety of poblem types. REFERENCE Aams, B.J., apa, F. (2000). Uban tomwate Management lanning with Analytical obabilistic Moels. John Wiley & ons, New Yok, UA. Benjamin, J.R., Conell, C.A. (1970). obability, tatistics, an Decision fo Civil Enginees, McGaw-Hill, New Yok, UA. Analytical obabilistic Moeling of Uban Dainage ystems age 15

16 Guo, Y., Aams, B.J. (1998a). Hyologic Analysis of Uban Catchments with Event-Base obabilistic Moels: 1. Runoff Volume. Wate Resouces Reseach 34(12), p Guo, Y., Aams, B.J. (1998b). Hyologic Analysis of Uban Catchments with Event-Base obabilistic Moels: 2. eak Dischage Rate. Wate Resouces Reseach 34(12), p Guo, Y., Aams, B.J. (1999a). Analysis of Detention ons fo tomwate Quality Contol. Wate Resouces Reseach 35(8), p Guo, Y., Aams, B.J. (1999b). An Analytical obabilistic Appoach to izing Floo Contol Detention Facilities. Wate Resouces Reseach 35(8), p Guo, Y., Aams, B.J. (1999c). The Analytical tomwate Management Moel (AWMM). In James, W. (e.) New Applications in Moelling Uban Wate ystems, Monogaph 7, p CHI, Guelph, Ontaio, Canaa. Kauffman, G.M. (1987). A Compaison of Analytical an imulation Moels fo Dainage ystem Design. M.A.c. Thesis, Dept. of Civil Engineeing, Univesity of Toonto, Toonto, Ontaio, Canaa. Li, J.Y., Aams, B.J. (2000). obabilistic Moels fo Analysis of Uban Runoff Contol ystems. Jounal of Envionmental Engineeing, ACE, 126(3), pp apa, F., Aams, B.J. (in pess 2002). Application of Analytical obabilistic Moels fo Uban Dainage ystems Analysis. New Tens in Wate an Envionmental Engineeing fo afety an Life: Eco-compatible olutions fo Aquatic Envionments; oceeings of the 2 n Intenational Confeence, Capi, June apa, F., Aams, B.J., Byant, G.J. (1997). Moels fo Wate Quality Contol by tomwate ons. In James, W. (e.) Avances in Moeling the Management of tomwate Impacts - Volume 5, pp CHI, Guelph, Ontaio, Canaa. eto, M.Y.K. (1984). Compaison of Altenative Deive obability Distibution Moels fo Uban tomwate Management. M.A.c. Thesis, Dept. of Civil Engineeing, Univesity of Toonto, Toonto, Ontaio, Canaa. U.. Amy Cops of Enginees (1974). toage, Teatment, Oveflow, Runoff Moel: TORM L2520, Hyologic Engineeing Cente, Davis, Califonia, UA. NOTATION A b b C C R D a E E q F i F X (x) catchment aea (ha) inteevent time (h) aveage inteevent time (h) aveage annual faction of pollution contolle aveage annual faction of unoff contolle aveage annual volume of epession stoage loss (mm) oveall T emoval efficiency in active stoage zone une ynamic settling oveall T emoval efficiency in pemanent pool une quiescent settling faction of total mass containe in i th size faction CDF of the continuous anom vaiable X Analytical obabilistic Moeling of Uban Dainage ystems age 16

17 f X (x) DF of the continuous anom vaiable X G (0) pobability pe ainfall event that any magnitue of spill fom stoage will occu G (p o ) pobability pe ainfall event that the volume of spill equals o excees p o G X (x) complementay CDF of the continuous anom vaiable X h A epth of active stoage zone (m) h epth of pemanent pool (m) i aveage intensity of ainfall event (mm/h) i aveage of aveage intensity of ainfall events (mm/h) i e unoff intensity (mm/h) L aveage annual loss volume (mm) n p aveage annual numbe of unoff events which o not utilize stoage n aveage annual numbe of unoff events n s aveage annual numbe of spills p volume of unoff spille fom esevoi (mm) u aveage annual volume of spills (mm) w aveage annual volume of unoff pocesse without stoage (mm) R aveage annual volume of unoff (mm) R aveage annual faction of unoff spille A maximum (active) stoage volume of ownsteam esevoi (mm) epession stoage (mm) s i volume of contents in stoage at the en of the i th (pevious) event (mm) stoage capacity of a pemanent pool o of a stoage esevoi with no outlet (mm) t uation of ainfall event (h) t aveage uation of ainfall events (h) T faction of pollution emoval efficiency eceive by spills oute though stoage t awown time of esevoi (h) t uation of unoff event (h) t aveage uation of unoff event (h) T R etun peio (y) t s aveage steay-state etention time of the pon (h) v total volume of ainfall event (mm) v aveage total volume of ainfall events (mm) v volume of unoff (mm) V teminal settling velocity of a iscete paticle size (m/h) V i aveage settling velocity of paticles in the i th size faction (m/h) β paamete fo exponential DF of aveage ainfall intensity (h/mm) γ paamete fo geneic exponential DF paamete fo exponential DF of ainfall volume (mm -1 ) η pollution emoval efficiency of unoff contolle by stoage/teatment system η pollution emoval efficiency of stoage facility pollution emoval efficiency of teatment facility η Analytical obabilistic Moeling of Uban Dainage ystems age 17

18 λ paamete fo exponential DF of ainfall uation (h -1 ) θ aveage annual numbe of ainfall events Φ ate of loss of epession stoage aveage acoss stom uation (mm/h) unoff coefficient ψ paamete fo exponential DF of inteevent time (h -1 ) contolle constant outflow ate of esevoi (mm/h) Analytical obabilistic Moeling of Uban Dainage ystems age 18