LINEAR-QUADRATIC REGULATORS APPLIED TO SEWER NETWORK FLOW CONTROL

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1 LIEAR-QUADRATIC REGULATORS APPLIED TO SEWER ETWORK FLOW COTROL Magdalene Marnak and Markos Papageorgou Dynamc Systems and Smulaton Laboratory, Department of Producton Engneerng and Management, Techncal Unversty of Crete, Unversty Campus, Chana, GREECE E-mal: Fax: Keywords: Water network control, Multvarable regulators, Feedforward terms, Lnear-Quadratc regulator. Abstract The problem consdered addresses the optmal wastewater dstrbuton to several retenton reservors n an urban sewer network durng ranfall n the am of protectng the qualty of recevng waters va mnmzaton of overflows. To ths end, a lnear multvarable feedback regulator s developed usng the lnear-quadratc desgn procedure. Inflow predctons are accommodated va sutable n the control law. A study for a real large-scale combned sewer network usng ths method s presented on the bass of a realstc smulaton model. Results demonstrate the effcency of the developed methodology. 1 Introducton The constructon of treatment plants for sewage protects the qualty of the waters that receve the outflows of the sewer networks. However, urban combned sewer networks do not have separated collectors for the domestc and ndustral sewage and the ranwater dranage. Therefore, durng ranfall, networks and/or treatment plants may be overloaded, and overflows of untreated wastewater may take place upstream of overloaded stretches, causng polluton of recevng waters. Placng retenton reservors at approprate locatons of the network s a cost-effcent way to avod overflows at moderate ran events and to reduce them at stronger ranfall as the water s stored n the reservors durng the ranfall and s drected towards the treatment plant after the ranfall stops. Optmal operaton of the combned sewer network (that contans retenton reservors) mples that for each ran event the whole retenton capacty of all reservors wll be used before overflows take place somewhere n the network. Ths, however, cannot be guaranteed by fxed gate settngs, such as fxed wers or manually adjustable gates for the fllng and emptyng of the storage spaces. Especally f the ranfall s dstrbuted unevenly over the urban area, there may be reservors that are not totally flled, whle overflows occur elsewhere n the network. In these cases, a further reducton of overflows can be obtaned va real-tme operaton of the reservors, e.g. by use of controllable gates that are drven by an automatc control strategy. On the other hand, an effcent control strategy may lead to substantal cost savngs, as the number and storage capactes of the reservors requred to keep overflows below a certan (usually legslatvely defned) lmt, depends upon the effcency of the appled control strategy. A real-tme control structure for sewer networks that combnes hgh effcency and low mplementaton cost, may be composed of a number of control layers (multlayer control structure). Such a flexble, relable, and effcent herarchcal control structure for real-tme control of sewer networks has been proposed e.g. n [13]: An adaptaton layer s responsble for ran and/or nflow predcton (f needed) and for real-tme estmaton of the system state. An optmzaton layer s responsble for the central, overall network control,.e. for specfyng reference trajectores for the reservor storages and outflows. A decentralzed drect control layer s responsble for the realzaton of the reference trajectores. Wth regard to the optmzaton layer, several approaches have been proposed n the past, lke: onlnear optmal control [1, 4, 9, 5, 14]. Multvarable feedback control [8, 4, 6]. Methods based on dynamc [] or lnear programmng [10]. Expert systems, fuzzy control [1], and further heurstc approaches. Ths paper focuses on the multvarable feedback control approach to central control of sewer network flow. Several mprovements, modfcatons and extensons ntroduced to prevously developed versons of the method [8] n order to ncrease ts effcency are ncluded. The man paper focus compared to prevous studes [6] s on applcaton and testng of ths method by use of a realstc smulaton model, the program KASIM. Control Problem Formulaton.1 Mathematcal model and constrants For the study of the sewer network control problem, two mathematcal models of the sewer network are employed, a realstc smulaton model (KASIM) and a smpler control desgn model that s referred to as smplfed model. The smplfed model s used for the desgn of the multvarable regulator whle KASIM [7] s used for testng the performance and senstvty of the control method. Wthn

2 KASIM all the dynamc phenomena that take place n the dfferent elements of the sewer network are modeled n detal usng known laws of hydraulcs such as the Sant-enant equatons for the sewer stretches, the Polen formula for overflows, etc., whle the smplfed model has lower accuracy and complexty. Combned sewer networks consst of a set of elements n whch dfferent processes take place, as for example storage (n the reservors or n the sewers), transport (n the sewers), mergng of flows (n the nodes). Τhe typcal elements upon whch a combned sewer network may be bult accordng to the smplfed model [3, 6] are reservors, nodes, external nflows, lnk elements and treatment plants (Fgure 1). A partcular network can be assembled from these elements. The whole flow process may be consdered to have a vector nput u ncludng all controllable reservor outflows, a dsturbance vector d ncludng all external nflows, and a state vector x ncludng all reservor storages and lnk outflows. Then, the model equatons may be expressed n the followng general form [3] x(k+1) = f[x(k), x(k 1),...,x(k κ x ), u(k), u(k 1),...,u(k κ u ), d(k), d(k 1),...,d(k κ d )] (1) where k= 0,1,... s the dscrete tme ndex and κ x, κ u, κ d are the longest tme delays of x, u and d, respectvely. The control constrants have the form u mn u(k) u max (x(k),k) κ (a) u Α(k) u(k κ) c( x(k), k) (b) κ= 1 where A(k) s a matrx of zero/one coeffcents. In contrast to KASIM, the smplfed model does not consder backwater effects.. Control Objectves The man task of the control system s the mnmzaton of overflows for any ranfall event. Ths can be acheved by: Usng all avalable storage space before allowng an overflow to occur somewhere n the network. Moreover, f, due to strong ranfall, overflows are unavodable, they should be dstrbuted as homogeneously as possble over tme and over the network reservors. However, f there are storage elements wthout overflow capablty (no overflow wers), the avodance of overloadng of these storage elements s of even hgher mportance. Emptyng the network as soon as possble (by fully usng the nflow capacty of the treatment plant) so as to provde free storage space for a possble future ranfall. A drect way of consderng these man objectves, along wth some secondary operatonal objectves, s va mnmzaton of a nonlnear objectve functon [3]. In ths paper, an alternatve approach that leads to a quadratc objectve crteron s taken [3]. 3 Regulator Desgn 3.1 Lnear-quadratc formulaton For the sewer network flow control problem, applcaton of the lnear-quadratc-regulator (LQR) methodology appears most convenent [8]. The LQR methodology s not drectly applcable n presence of tme delays, lke those appearng n the process model (1). Ths dffculty may be readly crcumvented by ntroducng some auxlary varables x [6]. Thus, f a control varable uj appears n the model equatons wth tme delay ntroduce the addtonal auxlary state equatons and substtute x κ uj, one may x1 (k + 1) = u j (k) x (k + 1) = x1(k) (3) x (k + 1) = x (k) κuj κuj κuj-1 (k) n all model equatons where u j (k κ uj ) appears. Ths modfcaton can be performed for all tmedelayed control and state varables of the process model. The auxlary varables x are regarded as addtonal state varables that are ncorporated n the state vector x. Wth ths modfcaton, (1) obtans the smpler form x(k+1) = f[x(k), u(k),d(k)]. (4) To facltate the applcaton of LQR desgn, lnearzaton around a statonary nomnal pont s requred. For the defnton of ths pont, a nomnal ranfall s consdered that leads to constant nomnal external nflow values d such that d 1 + d d nd = r max (r max s the plant s maxmum capacty) results n absence of any control actons (all gates opened). Under nomnal condtons no overflows occur, because we have assumed that the sum of external nflows equals r max. Usng these values, the nomnal values for each reservor s outflow, the nomnal reservor storages, u q the nomnal values of the lnk outflows and of the auxlary varables x are obtaned [3]. The nomnal steadystate just descrbed corresponds to a steady-state form of (4) x Ν = f(x,u,d ). (5) Lnearzaton of (4) around ths steady-state leads to f x(k+1)= x f x(k)+ u f u(k)+ d(k) (6) d

3 where x(k)=x(k) x Ν, u(k)=u(k) u, and d(k)=d(k) d are the lnearzed varables, and A= f/ x, B= f/ u, C= f/ d are the state, control, and dsturbance matrces, respectvely, of the lnearzed system. For reasons not detaled here, the orgnal system (6) s not fully controllable [3]. To obtan a fully controllable lnear model, the n x state varables and accordng state equatons correspondng to the n x reservors, are replaced by n x 1 new state varables and state equatons. The new state varables and state equatons are obtaned by buldng n x 1 ndependent dfferences of the old state equatons. For example, f the lnearzed conservaton equatons of reservors and j are (k+1) = (k) T[ u (k) + d (k)] (7) j (k+1) = j (k) T[ u j (k) + u l (k)] (8) respectvely, where (k) s the storage n reservor and T s the dscrete tme nterval, a new state equaton may be obtaned wth new state varable x (k+1)= (k + 1),max (k + 1) j j,max Although other schemes may be envsaged, we consder a specfc reference reservor o. j n (9) whle =1,,n x, j. ote that the modfcaton (9) s appled only to the reservor state equatons, whle the other state equatons (for the lnk outflows and for the auxlary varables) reman unchanged. A quadratc crteron that consders the control objectves mentoned prevously has the general form (for smplcty x s used n the followng to denote x ) J= ( x(k) Q + u(k) R ) (10) k=0 where η S =η Τ Sη whle Q and R are nonnegatve defnte, dagonal weghtng matrces. The nfnte tme horzon n (10) s taken n order to obtan a tme-nvarant feedback law accordng to the LQ optmzaton theory [11]. Due to the defnton of x(k), the frst term n (10) penalzes relatve storage dfferences between reservors. The dagonal elements of Q correspondng to the reservor storages x are set equal to 1, whle the dagonal elements of Q correspondng to the lnk outflows q and those correspondng to the auxlary varables x are set equal to zero. A controller desgned to mnmze ths crteron, wll automatcally tend to equalze the relatve storage dstrbuton between reservors. Ths s an ndrect way of achevng overflow mnmzaton for the sewer network. By the choce of the weghtng matrx R,.e. ts dagonal elements, the magntude of the control reactons can be nfluenced. Ths s necessary n order to avod hgh feedback j (9) parameters that would lead to nervous control behavor. Moreover, t provdes the possblty to consder, to a certan extent, ndrectly, the constrants (), because ncreased values of the dagonal weghtng parameters wll lead to lower devatons of outflows from ther nomnal values. However, the strct consderaton of the constrants () s not guaranteed by the quadratc crteron and must be mposed after the feedback law calculatons,.e. the control varables must be truncated accordng to (). The choce of the dagonal matrx R s performed by a tral-and-error procedure so as to acheve a satsfactory control behavor for a gven applcaton network. The nflow r(k) nto the treatment plant s not ncluded n the control vector u, but s set r(k) = r max. 3. Multvarable control law Two multvarable controllers, one wth and another wthout, were desgned va the LQR methodology n order to nvestgate both the reactve and antcpatory regulator behavor. The mnmzaton of the performance crteron (10) subject to the lnearzed state equaton, when nflow predctons are avalable ( d(k) 0), leads to the control law u(k)=u L x(k) U(k) (11) where L s a constant feedback gan matrx calculated from the well-known Rccat equaton. The tme-varant (feedforward) vector U(k) s calculated n real tme usng at each tme nstant k the predctons d(κ), κ=k,,k+k 1, where K s the predcton horzon. In the case of sewer network control, K corresponds to the horzon of the real-tme avalable nflow predctons K p (taken from a predctve ranfall-runoff model) plus the nflow predctons obtaned by the use of a smple extrapolaton scheme [3]. In the present study, the extrapolaton scheme uses the known values of the last three tme ntervals K p 1, K p, K p 3 to predct, usng lnear regresson, the nflow values for the next 0 mn after whch the nflows are assumed to move towards dry weather flow values, whch they reach 0 mn later. If d(k)=0, the tme-varant vector U(k) vanshes leadng to a purely feedback control law n (11). The state feedback regulator (11) requres avalablty of measurements for all state varables n real tme. In the sewer network context, measurements are typcally avalable for the reservor storages and possbly for some lnk outflows, but not necessarly for the retarded auxlary varables. Thus, f full real-tme measurements are lackng, some sort of state estmator may have to be developed n order to estmate the mssng measurements n real tme [3]. 4 Applcaton Example 4.1 Applcaton etwork To assess the effcency of the descrbed methodology n reducng the overflows and more generally n satsfyng the

4 control objectves when appled to a real sewer network, an extended nvestgaton was performed for the sewer network of Obere Iller (Bavara, Germany). Ths network connects fve neghborng ctes to one sngle treatment plant. The network has very long sewer stretches wth accordngly long flowng tmes n the lnk elements. The smplfed model of ths network s depcted n Fgure 1 whereby reservor 7 s a storage element created by nstallng a control gate to regulate the flow at the end of a volumnous sewer n the network wthout overflow capablty. There s, however, for emergency needs, a bypass of the control gate (a wer over the gate), so that n case of an overload, an overflow q over,7 s created that enters the sewer 5 through nodes 4 and 5. For the smplfed model of ths sewer network the equatons (1) are used. The dscrete tme nterval T s taken equal to 180 s for the control and 60 s for the KASIM-smulaton. The treatment plant has maxmum capacty r max = m 3 /s. For the LQ formulaton of the present problem, the auxlary varables (3) are used n order to take nto account the tme delays, and the correspondng equatons are added to the state equatons. Thus, takng nto account the tme delays, we have 159 state varables for the present problem (10 for the reservors, 6 for the lnk outflows, and 143 for the auxlary varables) [3]. arous scenaros of external nflows [3] were used n order to nvestgate the effcacy of the multvarable regulator for the partcular network under dfferent crcumstances. The control results for one of these scenaros, whch has farly nhomogeneous external nflows, are presented n ths paper. 4. Appled methodologes The multvarable controllers were programmed and were connected as an addtonal module to the smulaton program KASIM. For the desgn procedure, reservor 7 of the partcular applcaton network s consdered as the reference reservor j as ths reservor s geographcally n the center of ths sewer network. Thus we have as state varables x = ( )/(,max ) ( 7 7 )/( 7,max 7 ), =1,...,11, 7. After the calculaton of the control varables from (11) for the LQ regulators wth or wthout, a water level control scheme s actvated f necessary to keep the water level n reservor 7 near the value h w,7 (heght of the overflow wer of reservor 7) by approprate operaton of the control gates of the reservors upstream of reservor 7 and by ths way to avod the overloadng of ths storage element [3]. KASIM also smulates the underlyng actons of local drect control [7] wthn the multlayer control structure. Ths program s also used to smulate the no-control case, so as to llustrate the achevable mprovements va applcaton of an effcent central control strategy to the partcular network. In the no-control case, the gates are assumed opened to 8%, 7%, 100%, 50%, and 100% of ther maxmum openng heght for reservors 1, -6, 7, 8-10, and 11, respectvely. The selecton of the above percentages for the openng heghts of the orfces, whch have dfferent geometrc characterstcs, was performed by dong many smulaton nvestgatons usng dfferent percentages so as to acheve acceptable fxed-control performance wthout overloadng reservor 7. 5 Results 5.1 o-control case The smulaton results of the no-control case are summarzed n Table 1. In the scenaro consdered here whch has a duraton of 6 h, external nflows are stronger downstream of reservor 7 than upstream of reservor 7. Reservors 10, 8, and 9 receve very strong external nflows (d 11, d 8, d 10, respectvely) and thus, large overflows appear n these reservors (Table 1, Fgure a) when no control actons are taken. Reservors 1 and are also overflowng, although they do not have very strong external nflows. Ths s due to the openng heght of the gates of reservors 1 and. However, t should be noted that the selecton of the percentages of the openng heghts of the orfces leads to the avodance of overloadng of reservor 7 for ths partcular scenaro [3]. 5. Multvarable regulator wthout The man observatons n the scenaro presented here are summarzed n the followng remarks: The regulator manages to sgnfcantly reduce the total overflows n the network (Table 1) compared to the nocontrol case. Durng the crtcal perod where overflows occur, the regulator closes reservor 7 n order to equalze ts relatve storage wth that of the other reservors and so, reservors 8, 9 and 10, that are strongly overflowng n the no-control case, can have hgh outflows whch leads to a sgnfcant reducton of ther overflows (Fgure b). At the same tme, the overflows of reservors 1, are completely avoded (Table 1). The treatment plant s fed wth ts maxmum capacty, and so the network s empted as soon as possble n order to have free storage space for a possble future ranfall. 5.3 Multvarable regulator wth The multvarable regulator wth addtonal antcpates to some extent the mpact of future nflows. For the varous scenaros of external nflows used n [3] the results obtaned usng the regulator wth are equally effcent or slghtly superor to the control results obtaned usng the regulator wthout, dependng upon the partcular nflow event. When accurate nflow predctons are assumed avalable for the whole smulaton horzon, the results of Table 1 are obtaned for the scenaro consdered here. The regulator wth results n slghtly less overflows (Table 1) than the regulator wthout. Ths s because the regulator wth, knowng about the large nflow peaks that wll reach reservors 8, 9 and

5 d 1 d 3 d 10 d 11 1 q over,1 q over, d 4 4 q over, q over,10 q over,11 q over,7 d u 1 1 u out,1 q 1 u q u out, 1 u u 9 q u out,4 u u out, u 6 u out, 5 q u u 5 out, 6 8 u 3 u q 6 11 r q over,6 d 9 8 q over,8 d 1 u n, Legend q over,3 q over,4 d 13 d 5 d 6 q 3 3 d 8 q over, Reservor element u 5 u 5 q over,5 u n,,1 u n,, ode d 7 d q u, u out, r q External nflow Lnk element Treatment plant Fgure 1: Applcaton network. 10, retans more water n reservor 7, durng a certan crtcal perod, and thus, reservors 9 and 10 can have greater outflows and smaller overflows than the ones n the case wthout [3]. The mpact of naccurate nflow predctons on the regulator s behavor s also nvestgated. Thus, the regulator wth s appled when accurate nflow predctons are only avalable for 60 mn (K p =0), 30 mn (K p =10) or when there are no avalable predctons (K p = 0, that s we have only extrapolaton of current and past nflow values). The results are summarzed n Table 1 and they can be seen to be very smlar to the ones obtaned wth accurate nflow predcton. However, when only past values (a) (b) Fgure : Relatve reservor storages ( (k)/,max )100% for =7, 11, for the no-control case (a) and for the regulator wthout (b).

6 Reservor o control Regulator wthout Regulator wth K p =K Regulator wth K p =0 Regulator wth K p =10 Regulator wth K p = Total Table 1: Reservor overflows and overload of reservor 7 n [m 3 ]. are used for the predcton (K p =0) and, thus, an underestmaton or overestmaton of the future nflow values s more lkely, the results obtaned may not always be as good as the ones obtaned wth accurate nflow predctons. Indeed for K p =0, results are slghtly nferor to the ones of the regulator wth and accurate nflow predctons but are qute smlar to the ones of the regulator wthout. 6 Conclusons A generc problem for central sewer network control has been outlned. The developed methods, multvarable feedback regulator wth and wthout, have been appled to a large real-lfe sewer network. A realstc smulaton model has been used as a representaton of the real network for the assessment of the control results. The results obtaned were very satsfactory and were sgnfcantly better than the ones obtaned when no control actons were taken. These results ndcate that the man goals requred by a control system for combned sewer networks are met by these control methods. Acknowledgement Ths research was partally supported by the EC Research Project WWT & SYSEG Gettng Systems Engneerng nto Regonal Wastewater Treatment Strateges. References [1] L. Fuchs, T. Beeneken, P. Spönemann, and C. Scheffer. Model Based Real-Tme Control of Sewer System Usng Fuzzy-Logc, Water Scence Technology, ol. 36, o. 8-9, pp , (1997). [] J. W. Labade, D. M. Morrow, and Y. H. Chen. Optmal Control of Unsteady Combned Sewer Flow, Journal of the Water Resources Plannng and Management Dvson, Proceedngs of the ASCE, ol. 106, pp. 05-3, (1980). [3] M. Marnak. Optmal Real-Tme Control of Sewer etworks. PhD Thess, Techncal Unversty of Crete, Chana, Greece, (00). [4] M. Marnak, and M. Papageorgou. Central Flow Control n Sewer etworks, ASCE Journal of Water Resources Plannng and Management, ol. 13, o. 5, pp , (1997). [5] M. Marnak, and M. Papageorgou. A on-lnear Optmal Control Approach to Central Sewer etwork Flow Control, Internatonal Journal of Control, ol. 7, o. 5, pp , (1999). [6] M. Marnak, M. Papageorgou, and A. Messmer. Multvarable Regulator Approach to Sewer etwork Flow Control. ASCE Journal of Envronmental Engneerng, ol. 15, o. 3, pp , (1999). [7] A. Messmer. KASIM Documentaton. IB Messmer, Seeshaupt, Germany, (1998). [8] A. Messmer, and M. Papageorgou. Multreservor Sewer-etwork Control va Multvarable Feedback. ASCE Journal of Water Resources Plannng and Management, ol. 118, o. 6, pp , (199). [9] J.-F. Méthot, and M. Pleau. The Effects of Uncertantes on the Control Performance of Sewer etworks, Water Scence Technology, ol. 36, o. 5, pp , (1997). [10] F. elen. A Model to Assess the Performance of Controlled Urban Dranage Systems, Water Scence Technology, ol. 9, o. 1-, pp , (1994). [11] M. Papageorgou. Optmerung. (Second Edton). Oldenbourg erlag, Munch, Germany, (1996). [1] M. Papageorgou, and R. Mayr. Optmal Real-Tme Control of Combned Sewer etworks, IFAC Conference on Systems Analyss Appled to Water and Related Land Resources, Lsbon, Portugal, III-17-III-, (1985). [13] M. Papageorgou, and F. Mevus. Herarchcal Control Strategy for Combned Sewer etworks. Instrumentaton and Control of Water and Wastewater Treatment and Transport Systems, (R. A. R. Drake. (Ed)), pp Pergamon Press, Oxford, U.K, (1985). [14] M. Pleau, G. Pelleter, H. Colas, P. Lavallée, and R. Bonn. Global Predctve Real-Tme Control of Quebec Urban Communty s Westerly Sewer etwork, Water Scence and Technology, ol. 43, o. 7, pp , (001).