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Piece-wise linear functions-based modelpredictive control of large-scale sewagesystemsC. Ocampo-Martinez V. PuigInstitut de Robotica i Informatica Industrial (CSIC – UPC), Llorens i Artigas, 4-6, 08028 Barcelona, SpainE-mail: cocampo@iri.upc.edu

Abstract: In this study, model predictive control (MPC) of large-scale sewage systems is addressed, consideringseveral inherent continuous/discrete phenomena (overflows in sewers and tanks) and elements (weirs) in thesystem. This fact results in distinct behaviour depending on the dynamic state (flow/volume) of the network.These behaviours cannot be neglected nor can be modelled by a pure linear representation. In order to takeinto account these phenomena and elements in the design of the control strategy, a modelling approachbased on piece-wise linear functions (PWLF) is proposed and compared against a hybrid modelling approachpreviously suggested by the authors. Control performance results and associated computation times of theclosed-loop scheme considering both modelling approaches are compared by using a real case study based onthe Barcelona sewer network. Results have shown an important reduction in the computation time when thePWLF-based model is used, with an acceptable suboptimality level in the closed-loop system performance.

1 IntroductionSewer networks are considered as complex large-scale systemssince they are geographically distributed and interconnectedwith a hierarchical structure. Each subsystem is composedof a large number of elements with time-varying behaviour,exhibiting numerous operating modes and subject todynamic changes because of external conditions (weather)and operational constraints.

Most cities around the world have sewage systems thatcombine sanitary and storm water flows within the samenetwork. This is why these networks are known ascombined sewage systems (CSS). During rain storms,wastewater flows can easily overload these CSS, therebycausing operators to dump the excess of water into thenearest receiver environment (rivers, streams or sea). Thisdischarge to the environment, known as combined sewageoverflow (CSO), contains biological and chemicalcontaminants creating a major environmental and publichealth hazard. Environmental protection agencies havestarted forcing municipalities to find solutions in order to

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avoid those CSO events. A possible solution to the CSOproblem would be to enhance existing sewer infrastructureby increasing the capacity of the wastewater treatmentplants (WWTP) and by building new undergroundretention tanks. But in order to take profit of theseexpensive infrastructures, it is also necessary a highlysophisticated real-time control (RTC) scheme, whichensures that high performance can be achieved andmaintained under adverse meteorological conditions [1].The advantage of RTC applied to sewer networks has beendemonstrated by an important number of researchersduring the last decades, see [2–5].

An RTC scheme in sewage systems might be ‘local’ (whenflow regulation devices use only measurements taken at theirspecific locations) or ‘global’ (when control actions arecomputed taking into account real-time measurements allthrough the network). Since a sewer network as a complexsystem shows strong cross-relation between its elements, aglobal RTC is the proper strategy to manage and controlthis type of systems. The multivariable and large-scalenature of sewer networks have lead to the use of some

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variants of receding horizon control (RHC) – also referred asmodel predictive control (MPC) – as global control strategies[2, 5]. In order to use RHC within a global RTC scheme of asewage system, a model able to predict its future states over aprediction horizon taking into account a rain forecast isneeded.

Sewer networks are systems with complex dynamics sincewater flows through sewers in open channel. Whendeveloping a control-oriented model, there is always atrade-off between model description accuracy andcomputational complexity. Several control-orientedmodelling techniques presented in the literature deal withthe global RTC of sewage systems, see [5, 6]. In [7, 8], aconceptual linear model is used based on the assumptionthat a set of sewers in a catchment can be considered as a‘virtual tank’. The main reason to use a linear model is topreserve the convexity of the optimisation problems relatedto the RHC strategy. A similar approach can be found inan early reference on RHC applied to sewage systems [2].However, there exist several inherent phenomena (overflowsin sewers and tanks) and elements (weirs) in the systemthat result in distinct behaviour depending on the state(flow/volume) of the network. These discontinuousbehaviours cannot be neglected nor can be modelled by apure linear model. Additionally, the presence of intenseprecipitation causes that new flow paths appear. Thus,some flow paths are not always present in the sewernetwork and depend on its state and disturbances (rain).The description and analysis of these continuous/discretedynamic behaviours on sewer networks have beenpreviously reported by the authors [9]. In that work, it wasalso presented an hybrid modelling approach based on the‘mixed logical dynamical’ (MLD) form – introduced in[10] – oriented to the design of RHC-based RTC schemefor large-scale sewage systems. However, it was shown thatthe inclusion of those discontinuous behaviours in theRHC problem increases the computation time of thecontrol law. So, some relaxation in the modelling approachshould be thought such that it can be considered withinthe RTC of large-scale sewer networks.

The aim of this paper is to propose an alternative modellingapproach that represents the sewage system by using piece-wiselinear functions (PWLF; in the sequel called PWLF-basedmodel or simply PWLF model), following the ideas proposedby Schechter [11]. The purpose of this modelling approachis to reduce the complexity of the RHC problem by avoidingthe logical variables introduced by the MLD systemrepresentation. The idea behind the PWLF-based modellingapproach consists in having a description of the networkusing functions that, despite their discontinuous nature, areconsidered as quasi-convex [12], fact that might yield to thequasi-convexity of optimisation problems associated to thenon-linear MPC strategy used for RTC of the sewagesystem [8]. In this way, the resultant optimisation problemsdo not include integer variables, what allows savingcomputation time.

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The remainder of the paper is organised as follows. InSection 2, control-oriented modelling of sewer networks isrevised. The issue of hybrid dynamics is presented andaddressed using the proposed PWLF-based modellingapproach. RTC scheme for sewage systems based on MPCstrategy is addressed in Section 3, taking into account themodelling approach presented in the previous section.Section 4 presents the real case study based on theBarcelona sewer network used in this paper. This casestudy is used to show the computational improvementswhen implementing a predictive controller based on themodelling approach presented in Section 2 with respect tothe hybrid approach suggested in [13]. Section 5 showsand discusses the control results obtained in severalscenarios in the proposed case study. Finally, in Section 6the conclusions are drawn.

2 RTC-oriented modelling oflarge-scale sewer networksOne of the most important stages on the design of RTCschemes for sewer networks, in the case of using a model-based control technique as MPC, lies on the modellingtask. This is because performance of model-based controltechniques relies on model quality. So, in order to designan MPC-based RTC scheme with a proper performance, asystem model with accuracy enough should be used butkeeping complexity manageable. This section reviews theprinciples of the mathematical RTC-oriented modelling ofsewer networks. Additionally, it shows how the proposedPWLF-based modelling framework is used to take intoaccount the inherent hybrid behaviours in the differentelements of the sewer network.

2.1 Principles of mathematical RTCmodelling of sewage systems

Water flow in sewer pipes is open-channel, whichcorresponds to the flow of a certain fluid in a channel inwhich the fluid shares a free surface with an empty spaceabove. The Saint–Venant equations, based on physicalprinciples of mass conservation and energy, allow theaccurate description of the open-channel flow in sewerpipes [14] and therefore also allow to have a detailed non-linear description of the system behaviour. These equationsare expressed as

∂qx,t

∂x+

∂Ax,t

∂t= 0 (1)

∂qx,t

∂t+ ∂

∂x

q2x,t

Ax,t

( )+ gAx,t

∂Lx,t

∂x− gAx,t(I0 − If ) = 0 (2)

where qx,t is the flow (m3/s), Ax,t is the cross-sectional area ofthe pipe (m2), t is the time variable (s), x is the spatial variablemeasured in the direction of the sewage flow (m), g is thegravity (m/s2), I0 is the sewer pipe slope (dimensionless), If

is the friction slope (dimensionless) and Lx,t is the water

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level inside the sewer pipe (m). This pair of partial-differential equations constitutes a non-linear hyperbolicsystem. For an arbitrary geometry of the sewer pipe, theseequations lack of an analytical solution. Notice that theseequations describe the system behaviour in high detail.However, such a level of detail is not useful for RTCimplementation because of the complexity and the highcomputational cost of combining (1) and (2) with theRHC strategy.

Alternatively, several conceptual modelling techniques thatdeal with RTC of sewer networks have been presented in theliterature, see [5, 6, 15, 16], among many others. Themodelling approaches presented in this paper follow closelythe mathematical modelling principles proposed in [2].Here sewage system is divided into catchments and the setof pipes storage capacity belonging to each partition ismodelled as a virtual tank. At any given time, the storedvolumes represent the amount of water stored inside thesewer pipes associated with. The virtual tank volume iscalculated through the mass balance of the stored volume,the inflows and outflows of the catchment measured usinglimnimetres and the input rain intensity measured usingrain-gauges.

Using the virtual tank modelling principle and the massbalance conservation law, a sewer network can bedecomposed and described by using the elementary modelsexplained below and shown in Figs. 1 and 2, element byelement and conforming a simple network, respectively.Other common sewage system elements such as pumpingstations can be easily represented by using the mentioned

Figure 1 Conceptual schemes for sewer networksconstitutive elements

a Virtual tankb Real tankc Redirection gated Sewer pipe or weir with single inflow

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modelling principles but will be omitted here as they arenot taken into account in the case study presented in thispaper. Next, sewer network constitutive elements will beexpressed including a conceptual scheme which will be notonly used for describing its operation but also forexplaining the mathematical relations and equations derivedwhen using the PWLF-based modelling approach.

2.2 PWLF-based modelling of sewernetwork constitutive elements

As discussed in the introduction, sewer networks presentseveral inherent hybrid behaviours that cannot be modelledusing a pure linear model. In this paper, the modellingframework based on PWLFs is used to model suchbehaviours. More precisely, the proposed PWLF-basedmodelling methodology consists in using continuous andmonotonic functions to represent expressions that containlogical conditions, which describe the weirs behaviour andoverflow capability of reservoirs, respectively. Indeed, thesephenomena involve the switching and discontinuousbehaviours of the sewage system. The PWLF approach isthough as an alternative to the use of a pure hybridmodelling approach, already proposed for the RTC ofsewer networks [13].

The PWL functions used to model the discontinuousbehaviours of sewer networks are defined as ‘saturation’ of avariable x in a value M (i.e. sat(x, m)), and ‘dead zone’ ofthe same variable x starting in a value M (i.e. dzn(x, M )).Those functions are monotonic and continuous and mightlead to a quasi-convex optimisation problem whenformulating the MPC problem. According to [12], theglobal optimal solution of quasi-convex optimisationproblems can be obtained by using a bisection method,which is logarithmic in time. This represents an advantagewith respect to the mixed-integer linear problems resultantwhen using a pure hybrid approach based on MLD orpiecewise affine (PWA) approaches. This type of modelsinduces an exponential complexity given by the handling ofBoolean variables and the discrete optimisation required.

2.2.1 Virtual and real tanks: These elements are usedas storage devices. In the case of virtual tanks (see Fig. 1a),the mass balance of the stored volume, the tank inflowsand outflows and the input rain intensity can be written asthe difference equation with a sampling time Dt

vik+1 = vik + DtwiSiPik + Dt(qinik − qout

ik) (3)

where vi corresponds to the volume in the ith tank at time k(given in cubic metres), wi is the ‘ground absorptioncoefficient’ of the ith catchment, S is the surface area andPk is the rain intensity at each sample k. Flows qin

ik and

qoutik are the sum of inflows and outflows, respectively. ‘Real

retention tanks, which correspond to the sewer networkreservoirs, are modelled in the same way but without theprecipitation term. Tanks are connected with flow paths or

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Figure 2 Simple sewer network composed of constitutive elements described in Section 2.2

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links, which represent the main sewage pipes between thetanks. Manipulated variables of the system, denoted as qui

,are related to the outflows from the control gates. Tankoutflows are assumed to be proportional to the tankvolume, that is

qoutik = bivik

(4)

where bi (given in s21) is defined as the ‘volume/flowconversion’ coefficient as suggested in [17] by using thelinear tank model approach. Notice that this relation can bemade more accurate (but more complex) if (4) is consideredto be non-linear (non-linear tank model approach). Limitson the volume range of real tanks are expressed as

0 ≤ vik≤ vi (5)

where vi denotes the maximum volume capacity given incubic metres. As this constraint is physical, it is impossibleto send more water to a real tank than it can store. Noticethat real tanks without overflow capability have beenconsidered. Virtual tanks do not have a physical upper limiton their capacity. When they rise above a pre-establishedvolume, an overflow situation occurs. This fact representsthe case when level in sewers has reached a limit so that anoverflow situation can occur in the streets (flooding).Hence, when virtual tanks maximum volume v is reached,the excess volume above this maximum amount isredirected to another tank (catchment) within the networkor to a receiver environment (as pollution). This situation

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creates a new flow path from the tank, denoted as qd

(referred to as ‘virtual tank overflow’) that can be expressedmathematically as

qdk=

(vk − v)

Dtif vk ≥ v

0 otherwise

⎧⎨⎩ (6)

In this case, the outflow of virtual tank is then limited by itsmaximum volume capacity as follows:

qoutk=

bv if vk ≥ v

bvk otherwise

{(7)

Consequently, considering the tank overflow, the differenceequation (3) in case of virtual tanks becomes

vik+1= vik

+ DtwiSiPik+ Dt(qin

ik − qoutk

− qdk) (8)

Using the proposed PWLF modelling approach, the tankoutflows can be expressed as

qoutk= bsat(vk, �v) (9)

qdk= dzn(vk, �v)

Dt(10)

On the other hand, as noticed before, real tanks (see Fig. 1b)are elements designed to retain water in the case of intenserain. For this reason, both tank inflows and outflows are

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controlled. In the same way, tank inflow is constrained by thecurrent volume within the real tank, by its maximum capacityand by the tank outflow. Since real tanks are consideredwithout overflow capabilities, inflow is pre-manipulated byusing a redirection gate (explained in Section 2.2.2), whatleads to include the management policy in the model of thereal tank. Proceeding in this way, the value of themanipulated flow qw

akis restricted to the maximum flow

condition in the input gate, and the flow through inputlink qa is expressed as

qak=

qwak

if qwak≤ qink

qink otherwise

{(11)

However, maximum tank capacity also constrains the inflowaccording to the expression

qak=

qakif qbk

− qoutk≤ v − vk

Dtv − vk

Dtotherwise

⎧⎪⎨⎪⎩ (12)

Thus, the real tank outflow is given by

qoutk= qw

outkif qw

outk≤ bvk

bvk otherwise

{(13)

taking into account that qwout is also limited by the maximum

capacity of the outflow link, denoted by qoutk, leading to the

following difference equation:

vk+1 = vk + Dt(qak− qoutk

) (14)

Notice that the flow through qb can be derived from the massbalance

qbk= qink

− qak (15)

The PWLF model for this element considering the tankinflow and outflow expressions is

qoutk= sat(qw

outk, bvk) (16)

qak= sat qw

ak, min

v − vk

Dt, qink

( )( )(17)

2.2.2 Manipulated gates: Within a sewer network,gates are elements used as control devices since they canchange the flow downstream. Depending on the actionmade, gates can be classified as ‘redirection gates’, used tochange the direction of the sewage flow, and ‘retentiongates’, used to retain the sewage flow in a certain networkpoint (sewer or reservoir). In the case of real tanks, aretention gate is present to control the outflow. Virtualtank outflows cannot be closed but can be diverted usingredirection gates. Indeed, redirection gates divert a flowfrom the nominal path which the flow follows if the gate isclosed. This nominal flow is denoted as Qi in the equation

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below, which expresses the mass conservation relation inthe element

qoutik = Qik

+∑

j

qjuik

(18)

where j is an index over all manipulated flows coming fromthe gate. Fig. 1c shows a conceptual scheme of redirectiongates considered in this paper. Assuming that the flowthrough sewer qa is imposed (e.g. computed by means of acontrol law), the expressions that describe a redirection gatecan be written as

qak= qa if qw

a > qa

qwa otherwise

{(19)

where qwa corresponds to the imposed/computed value for the

flow qak. Flow qbk

is directly given by the mass balanceexpression

qbk= qink

− qak(20)

In the case of redirection gates, the PWLF model is definedtaking into account that qa should satisfy the restriction (19),what can be rewritten in terms of the PWL functions as

qak= sat(qak

, qink) (21)

Flow through qb is given by the mass balance (20).

2.2.3 Weirs and main sewer pipes: Since thedescription of their dynamics is very close, both types ofelements are presented together in this section. ‘Nodes’ arepoints of the network where the sewage can be eitherpropagated or merged. Hence, these elements can be classifiedas ‘splitting nodes’ and ‘merging nodes’. The first type can betreated considering a constant partition of the sewage flow inpredefined portions according to the topological designcharacteristics. Merging nodes exhibit a switching behaviour.In the case of a set of n inflows qi, with i = 1, 2, . . . , n, theexpression for the node outflow qout is written as

qout =∑n

i=0

qi (22)

‘Weirs’ can be seen as splitting nodes having a maximum capacityin the nominal outflow path related to the flow capacity of theoutput pipe. In the same way, ‘main sewer pipes’ can be seenas weirs with a single inflow. They are used as connectiondevices between network constitutive elements. Thereforeconsidering the similarity between all the aforementionedelements and the notation in Fig. 1d, the set of expressionsthat represent the behaviour either of a weir or of a sewer pipe

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are the following:

qbk=

qb if qin . qb

qink otherwise

{(23)

qck=

qink− qb if qin . qb

0 otherwise

{(24)

where qb is the maximum flow through qb and qin is the inflow.Notice that the outflow from virtual tanks is assumed to beunlimited in order to guarantee a feasible solution of anassociated optimisation problem within the design procedureof a optimisation-based control strategy. The same idea appliesto the outflow qbk

related to retention gates. However, mostoften, sewer pipes have limited flow capacity. When the limitof flow capacity is exceeded, the resultant overflow is possiblyredirected to another element within the network or isconsidered as loss to the environment.

The PWLF model for main sewer pipes (or single inflowweirs) can be obtained from the overflow condition as follows:

qbk= sat(qink

, qb) (25)

qck= dzn(qink

, qb) (26)

where qb corresponds again to the maximum flow capacity ofthe nominal outflow pipe.

3 MPC-based RTC on large-scalesewer networks3.1 MPC as a tool for global RTC

In most sewer networks, the regulated elements (pumps,gates and retention tanks) are typically controlled locally,that is, they are controlled by a remote station according tothe measurements of sensors connected only to that station.However, a global RTC system requires the use of anoperational model of the network dynamics in order tocompute, ahead of time, optimal control strategies for thenetwork actuators based on the current state of the system[provided by supervisory control and data acquisition(SCADA) sensors], the current rain intensity measurementsand appropriate rainfall predictions. The computationprocedure of an optimal global control law should take intoaccount all the physical and operational constraints of thesewage system, producing set-points which achieveminimum flooding and CSO.

As discussed in the introduction, MPC is a suitablecontrol strategy to implement global RTC of sewernetworks since it has some features to deal with complexsystems such as sewer networks: big delays compensation,use of physical constraints, relatively simple for peoplewithout deep knowledge of control, multi-variable systemshandling etc. Hence, according to [1], such controllers arevery suitable to be used in the global control of urban

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drainage systems within a hierarchical control structure[18, 5]. MPC, as the global control law, determines theset-points for local controllers of the sewer network.A management level is used to provide MPC with theoperational objectives, what is reflected in the controllerdesign as the performance indexes to be minimised. In thecase of urban drainage systems, these indexes are usuallyrelated to flooding, pollution, control energy etc.

3.2 MPC on sewer networks

3.2.1 Control objectives: The sewage system controlproblem has multiple objectives with varying priority, see[5]. The type, number and priority of those objectives canalso be different depending on the particular sewage systemdesign. However, the most common objectives are generallyrelated to the manipulation of the sewage in order to avoidundesired sewage flows outside of the main sewers(flooding). The main considered objectives for the casestudy presented in this paper are listed below in order ofdecreasing priority:

† Objective 1: minimise flooding in streets (virtual tankoverflow).

† Objective 2: minimise flooding in links between virtualtanks.

† Objective 3: maximise sewage treatment.

A secondary purpose of the third objective is to reduce thevolume in the tanks to anticipate future rainstorms. Thisobjective also indirectly reduces pollution to theenvironment. This is because if the treatment plants areused optimally with the storage capacity of the network,pollution should be strongly minimised. Moreover, thisobjective can be complemented by conditioning minimumvolume in real tanks at the end of the prediction horizon.It could be seen as a fourth objective.

3.2.2 Problem constraints: When using the modellingrepresentations based on virtual tanks, either in MLD orPWLF form, only flow rates are manipulated in such waythat some the inherent non-linearities (e.g. non-linearrelation between gate opening and discharge flow) of thesewer network are simplified as discussed in [2]. However,in turn, some physical restrictions need to be included asconstraints on system variables. For instance, variables q j

ui

that redirect outflow from a virtual tank should never belarger than the outflow from the tank. This is expressedwith the following inequality:

∑j

qjuik

≤ qoutik = bvik

(27)

Additionally, operational constraints associated with therange of gates actuation lead to the manipulated flows haveto fulfill q

jui k

≤ q jui

, where q jui

denotes its upper limit.

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Similarly, operational limits on the range of real tank volumesshould be included (see (5)) to limit the amount of sewagethat can be stored.

3.2.3 MPC disturbances: Rain plays the role ofmeasured disturbance in the sewer networks MPCproblem. The type of disturbance model to be useddepends on the rain prediction procedure available.Existing methods range from the use of time series [19] tothe sophisticated utilisation of meteorological radars [20].According to [5], different assumptions can be done forthe rain prediction when an optimal/predictive control lawis used for the RTC of sewer networks. Results show thatthe assumption of constant rain over a short predictionhorizon gives results that can be compared with the thecase of considering known rain over the prediction horizon,confirming similar results reported in [2, 21].

4 Case study description4.1 Barcelona sewer network

The city of Barcelona has a CSS of approximately 1697 kmlength in the municipal area plus 335 km in themetropolitan area, but only 514.43 km are considered as themain sewer network. Its storage capacity is about threemillion of cubic metres, which implies a dimension threetimes greater than other cities comparable to Barcelona. It isworth to notice that Barcelona has a population which isaround 1.59 million inhabitants on a surface of 98 km2,approximately. This fact results in a very high density ofpopulation. Additionally, the yearly rainfall is not very high(600 mm/year), but it includes heavy storms (up to 90 mm/h)typical of the Mediterranean climate that can cause a lot offlooding problems and CSO to the receiving environments.

‘Clavegueram de Barcelona, SA’ (CLABSA) is thecompany in charge of the sewage system management inBarcelona. There is a remote control system in operationsince 1994 which includes sensors, regulators, remotestations, communications and a control centre in CLABSA.Nowadays, as regulators, the urban drainage system contains21 pumping stations, 36 gates, 10 valves and 8 retentiontanks which are regulated in order to prevent flooding andCSO. The remote control system is equipped with 56 remotestations including 23 rain-gauges and 136 water-level sensorswhich provide real-time information about rainfall andwater levels into the sewage system. All this information iscentralised at the CLABSA control centre through aSCADA system. The regulated elements (pumps, gates andretention tanks) are currently controlled locally, that is, theyare handled from the remote control centre according to themeasurements of sensors connected only to local stations.

4.2 Barcelona test catchment

This paper considers a portion of the Barcelona sewernetwork, which represents the main phenomena and the

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most common characteristics appeared in the entirenetwork. This representative portion is selected to be thecase study of this paper because a calibrated andvalidated model of the network obtained using thevirtual modelling methodology (see Section 2) is availableas well as rain gauge data for an interval of several years.The considered Barcelona test catchment (BTC) has asurface of 22.6 km2 and includes typical elements of thelarger network.

The BTC has one retention gate associated with one realtank, three redirection gates and one retention gate, 11sub-catchments defining equal number of virtual tanks,several level gauges (limnimetres) and two WWTPs. Also,there are five rain-gauges used to measure the rain enteringin each sub-catchment. Notice that some sub-catchments(virtual tanks) share the same rain sensor. These sensorscount the amount of tipping events in 5 min (samplingtime) and such values is multiplied by 1.2 mm/h in order toobtain the rain intensity P in m/s at each sampling time,after the appropriate units conversion. The differencebetween the rain inflows in virtual tanks that share thesame rain gauges is because of the particular surface area Si

and the ground absorption coefficient wi of thecorresponding subcatchment (see (3)), what yields todifferent amount of rain inflows.

Using the virtual tanks representation principle, theresultant BTC model has 12 state variables that representthe volumes of the 12 tanks (1 real and 11 virtual), 4control inputs that correspond to the manipulated gateflows and 5 measured disturbances associated to themeasurements of rain intensity at the sub-catchments. TwoWWTPs are used to treat the sewage before it is releasedto the environment. The states related to the virtual tankvolumes are estimated by using the limnimetres shown withcapital letter L in Fig. 3. The free flows to the environment(q10M, q7M, q8M and q11M to the Mediterranean sea andq12s to other catchment) and the flows to the WWTPs (q7L

and q11B) are also shown in the figure as well as rainintensities P13, P14, P16, P19 and P20. The four manipulatedflows, denoted as qui

, have a maximum capacity of 9.14, 25,7 and 29.3 m3/s, respectively. These limits cannot be relaxedsince they are physical restrictions of the system (hardconstraints).

4.3 Rain episodes

Rain episodes used for the simulation of the BTC and for thedesign of MPC strategies are based on real rain gauge dataobtained within the city of Barcelona on the given dates(yyyy-mm-dd) as presented in Table 1. These episodes wereselected because they represent the meteorological behaviourof Barcelona weather. Table 1 also shows the maximum‘return rate’ among all five rain gauges in each episode.(The return rate – or period – is defined as the averageinterval of time within which a hydrological event of givenmagnitude is expected to be equaled or exceeded exactly

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once. In general, this amount is given in years.) In thethird column of the table, the return rate for the wholeBarcelona network is shown. Notice that one of the rain

storms had a return rate of 4.3 years in the case of the wholenetwork while for one of the rain gauges the return rate was16.3 years.

Figure 3 Barcelona test catchment scheme

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5 Simulations and results5.1 Preliminaries

This section is focused on comparing the performance of theMPC-based sewer network RTC based on the PWLF-basedmodelling approach proposed in this paper with the hybridMLD modelling approach suggested in [9] using a set ofreal rain episodes in Table 1. Computation time, whenboth modelling approaches are used, is also compared.Results of such comparison would be a key issue to showthe benefits of using the modelling approach proposed inthis paper for RTC implementation in the real networkcase study. The assumptions made for all theimplementations will be presented and their validitydiscussed before the results are given. The detaileddescription of BTC case study including operating rangesof the control signals and state variables as well as thedescription of all variables and parameters can be found in[9, 13].

Remark 1: In order to give an adequate idea of thecomplexity of the hybrid problem, system model in MLDform can be equivalently represented in PWA form asstated in [22]. Hence, the system can be expressed as

x(k + 1) = Aix(k) + Biu(k) + fi , i = 1, . . . , Nd

forx(k)

u(k)

[ ][ Vi (28)

where x(k) [ Rnc is the vector of nc system states (networkvolumes at tanks), u(k) [ Rmic corresponds to the vectorof the mic control inputs (manipulated flows), fi arereal vectors associated to the model disturbances (rain) andNd denotes the number of different dynamics.

According to [23], the way of obtaining the PWA systemfrom the MLD form is based on the ennumeration of all 2rℓ

combinations of binary variables, where rℓ is the number ofauxiliary Boolean variables of the MLD model. In the caseof the application shown in this paper, the MLD modelconsiders rℓ = 22 (see [9] for further details), what leads in

Table 1 Rain episodes used for comparing modellingapproaches

Rain episode Maximum returnrate, years

Return rateaverage, years

1999-09-14 16.3 4.3

2002-07-31 8.3 1.0

2002-10-09 2.8 0.6

1999-10-17 1.2 0.7

2000-09-28 1.1 0.4

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Nd = 4 194 304 different dynamics. Since the complexityassociated to the dimension of the PWA model is highenough, the alternative of using this representation for theMPC design was not considered.

5.2 Simulation and prediction models

Results presented in this paper have been obtained insimulation by using two different models: one used as theplant (sewer network), which in the sequel will be called as‘open-loop model’, and the other used by the MPCcontroller, called ‘control model’. The open-loop model isimplemented considering a non-linear representation of thesewer network based on mass balances where ranges andbounds for every variable (control signals, volumes and raindisturbances) are strictly considered and all passible logicalor discontinuous dynamics are included (as the case ofweirs and overflows). On the other hand, the control modelis obtained by using the modelling approach presented inSection 2. This model, considering the PWLF-basedmodelling approach, was obtained by joining the differentcompositional elements described in Section 2.2 andfollowing the network diagram of Fig. 3, resulting in anon-linear representation as a set of expressions for thewhole network.

5.3 MPC controller set-up andimplementation

Different parameters of the MPC controller should bedefined and tuned according to the control objectives andtheir prioritisation. According to the discussion in Section3.2.1, to take into account the control objectives for theRTC of a sewer network, the following system outputshave been included in the control model

y1k=∑

i

qstrvk +∑

j

qstrqk (29a)

y2k=∑

l

qseak(29b)

y3k= qtrp1k

− q7L (29c)

y4k= qtrp2k

− q11B (29d)

where y1krepresents the sum of the i overflows to street from

virtual tanks at time k, denoted by qstrvk, plus the sum of the joverflows to street from links (main pipes) at time k, denotedby qstrqk. Output y2k

represents the sum of the l overflows that

are released to the sea (as receiver environment) at time k,denoted as qseavk and finally y3k

and y4krepresent the

difference at time k between the flows towards the WWTPs,denoted by qtrp1k

and qtrp2k, and the maximum WWTPs

inflows. For the case study of this paper, qtrp1k= q7Lk

andqtrp1k

= q11Bk, with their maximum flows q7L and q11B,

respectively.

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Using the outputs (29), the multi-objective cost functionfor the BTC can be written as follows using the weightedapproach technique

J (uk, xk) =∑Hp

i=1

‖yk+i|k‖2Q (30)

where yk+i|k is the output vector at the instant k + i withrespect to time instant k and Hp denotes the predictionhorizon set equal to 6, which is equivalent to 30 min witha sampling time Dt ¼ 300 s. This selection was based onthe reaction time of the system to disturbances. Anotherreason for this selection is that the constant rain predictionassumed in this paper becomes less reliable for largerhorizons. The length of the simulation scenarios is 100samples, what allows to see the influence of the peak of therain (disturbance) from the selected rain episode over thedynamics of the network and also over the dynamic of theclosed loop. Q corresponds to a matrix containing theweights wi, each one related to a control objective. Noticethat the desired prioritisation of the control objectives isgiven by the values wi that, for this case, determines a Qmatrix of the following form:

Q = diag{wstrI wseaI wtrp1I wtrp2I } (31)

where I corresponds to a identity matrix of suitabledimensions. Here, wstr = 1, wsea = 10−1, wtrp1 = 10−3 andwtrp2 = 10−3.

Notice that, considering the PWLF-based modellingapproach, a non-linear optimisation problem is stated. Byreplacing the non-linear equality constraints (coming fromthe definition of the PWLF model) in the objectivefunction (30), a new optimisation problem with PWLF-based objective function and bounded constraints isobtained. The selection of the algorithm to solve suchproblem was done after the evaluation of several solversavailable on Tomlabw (e.g. ‘conSolve’, ‘nlpSolve’, amongothers). The ‘structured trust region’ (STR) algorithm (see[24]) was finally chosen because it provides an ‘acceptable’trade-off between system performance and computationtime. If the global optimum is desired, the mentioned non-linear algorithm should be combined with a bisectionapproach as suggested in [12].

On the other hand, the MLD hybrid model of the BTC,suggested in [13], is used for the comparisons presentedbelow with the PWLF model. This MLD model has 22logical variables and 44 auxiliary variables. Using thismodel, a hybrid MPC controller has been designed and theset of considered rain scenarios were simulated using the‘Hybrid Toolbox’ for Matlabw (see [25]) and ILOG CPLEX

11.2. This latter solver allows to handle efficiently themixed-integer programming (MIP) problems associated tothe hybrid MPC controller.

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5.4 Control performance andcomputation time comparisons

For performance comparison purposes and additionally tothe control results when the considered modellingapproaches (hybrid and PWLF models) are used, resultsobtained when the network is in open loop are alsopresented. The computation times reported in this paperhas been obtained using Matlabw 7.2 implementationsrunning on an Intelw CoreTM 2, 2.4 GHz machine with4 Gb RAM. Notice that computation time results reportedhere related with hybrid models are different from thosepresented in [13] because of the machine characteristicsand solver versions.

The open-loop case consists in the sewage system withoutcontrol so the manipulated links are used as passive elements,that is, the amount of the flows qu1, qu2 and qu4 only dependon the inflow to the corresponding gate and they are notmanipulated while qu3 is the outflow of the real tank givenby gravity (tank discharge). Results related to the controlperformance are summarised in Tables 2–4 for fiverepresentative rain episodes in Barcelona between 1998 and2002 (yyyy-mm-dd in tables) presented in Table 1. Table 2shows the comparison of volumes of sewage released to thestreet (flooding) while Table 3 shows the same comparisonbut regarding the volumes to receiver environments(pollution). Finally, Table 4 shows the comparison ofvolumes related to the treated sewage at the WWTPs.

Notice from Tables 2–4 that the performance of the systemis better when an MPC control law is considered no matter the

Table 2 Performance results (flooding (×103 m3))

Rain episodes Open loop PWLF model Hybrid model

1999-09-14 108 88.2 92.9

2002-10-09 116.1 113.3 97

2002-07-31 160.3 132.8 139.7

1999-10-17 0 0 0

2000-09-28 1 1 1

Table 3 Performance results (pollution (×103 m3))

Rainepisodes

Openloop

PWLF model Hybridmodel

1999-09-14 225.8 226.1 (1.16%) 223.5

2002-10-09 409.8 407.7 (2.25%) 398.7

2002-07-31 378 380 (1.44%) 374.6

1999-10-17 65 59.9 (3.09%) 58.1

2000-09-28 104.5 102 (4.08%) 98

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modelling approach utilised with respect to the performance inopen loop. This justifies the use of closed-loop control.Moreover, notice also that the use of the hybrid modellingapproach implies in average a better system performancewith respect to the performance improvement obtained byusing the PWLF-based modelling approach. Notice alsothat the performance improvement is basically related to theimprovement of the main control objective, and thenfollowing in a hierarchical order, to the second objective andso on. In this way, notice that the pollution for someepisodes is worse with respect to the open-loop case – see,for example, Table 3, episodes 1999-09-14 and 2002-07-31.However, notice that the performance index associated tothe flooding is the best for both episodes, following the pre-established control objectives priority.

In general, these results were expected since the MPCcontroller based on the hybrid modelling approach achievesits optimum by solving a set of convex linear problemsusing a branch and bound scheme. However, the MPCbased on PWLF-based modelling approach leads to a non-linear network model representation what might result in aquasi-convex optimisation. Therefore using the STRalgorithm, the global optimum cannot be assured becausebisection method was not implemented in this paper. Thisfact leads possibly to a sequence of suboptimal controlactions when the computation of the RHC law is done.This explains why the performance obtained using thePWLF model is in general worse than the one obtainedusing the hybrid model. Suboptimality levels of the resultsobtained using the PWLF model were never greaterthat 4.1% for the cases of the second and third objective(as shown in Tables 3 and 4 in parenthesis). For the case ofthe first control objective (related to flooding), results werenot so homogeneous since for some scenarios one of themodelling approaches leads in better system performancewhile for other scenarios occurred just the opposite.

On the other hand, the main difference of using the hybridor the PWLF modelling approaches is in the computationtime required to determine the control actions at eachiteration. As mentioned in Section 5.2, the model in MLDform contains an important number of Boolean andauxiliary variables. In the BTC case study, to determine the

Table 4 Performance results (treated sewage at WWTPs(×103 m3))

Rainepisodes

Openloop

PWLF model Hybridmodel

1999-09-14 278.3 276.7 (1.43%) 280.7

2002-10-09 533.8 534.2 (1.98%) 545

2002-07-31 324.3 321.9 (1.80%) 327.8

1999-10-17 288.4 293.5 (0.61%) 295.3

2000-09-28 285.3 287.5 (1.51%) 291.9

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control actions using hybrid MPC implies that for eachtime instant, considering a prediction horizon Hp = 6,222×6 = 5.4 × 1039 linear programming (LP) problems (fora linear norm in the cost function) or quadraticprogramming (QP) problems (for a quadratic norm in thecost function) should be solved in the worst case. Thus, thecomplexity of the MIP associated to the MPC lawbecomes bigger by increasing the number of Booleanvariables since the underlying optimisation problem iscombinatorial and NP-hard [26]. Thus, the worst-casecomputation time is exponential in the amount of integervariables. In large-scale systems such as sewer networks, theamount of elements with logical/discontinuous dynamicscan augment according to the topology of the particularcase study. Therefore computation times increase towards apoint where the use of this modelling for obtaining anMPC-based RTC law becomes almost impossible. On theother hand, the alternative modelling approach based onthe PWLFs proposed on this paper allows to have controlsequences computed in lower times at the price of somedegree of suboptimality because of the possible localoptimum. Table 5 summarises the computation times forboth the modelling approaches proposed on this paper andfor the five rain episodes previously considered. It can beseen for the last two rain episodes that the computationtimes for both modelling approaches are almost equivalent.According to [9], this difference relies on the influence ofthe system disturbances within the optimisation problem.Depending on the variability of the rain and its intensity infunction of the location, more discontinuous behavioursmight be observed. This fact can determine the relativecomplexity of the MIP problem but it does not influencein the same way the optimisation problem based on PWLFmodels. When a hybrid model is used, depending on theevolution of the rain intensity, many modes can be changedand the global behaviour of the system represented by thehybrid model becomes more complex (e.g. new flows can

Table 5 Computation time results (s)

Rainepisodes

PWLF model Hybrid model

TotalCPUtime

MaximumCPU time in

a sample

TotalCPUtime

MaximumCPU time in

a sample

1999-09-14

695.33 91.32 1109.29 787.17

2002-10-09

293.23 66.01 561.73 85.31

2002-07-31

830.20 83.04 1050.54 381.49

1999-10-17

180.22 16.15 79.14 10.39

2000-09-28

120.88 12.13 84.76 13.27

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appear, Hamming distance between the Boolean variablesvectors in actual time instant and the previous step getsbigger etc.). This fact makes that the mixed-integeroptimisation gets also complex and requires highcomputational time. On the other hand, when a PWLF-based model is used under the same rain intensityconditions, despite the optimisation is non-linear, thecomputation does not depend on Boolean variables (whichdetermine the current mode of the system in the hybridmodel) and the computational time gets lower.

Summarising, although the suboptimal nature of thesolutions as a consequence the minor improvement of thecontrol performance, the MPC controller considering thePWLF-based modelling approach not only leads to a fastercontrol sequences computation but also to feasible oneswith respect to the real-time restriction imposed by thesampling time. In average, all the maximum computationtimes to compute the MPC control action when thePWLF-based modelling approach is used are less than thethird part of the sampling time. This is not the case whenusing the hybrid modelling approach.

6 ConclusionsIn this paper, RHC of large-scale sewage systems has beenaddressed considering a modelling approach based onPWLF. This modelling approach is compared against ahybrid modelling approach previously reported by the authorswithin the RHC framework. Control performance resultsand associated computation times of both approaches werecompared by using a real case study based on the Barcelonasewer network. With the PWLF-based modellingformulation proposed, although a small amount ofsuboptimality is introduced since the resultant non-linearoptimisation problem is non-convex, the reduction in thecomputation time allows to face the control of large-scalesewer networks. The future work is already focused on thequasi-convexity theoretical proof of optimisation problembased on PWLF models in order to take advantage of themodelling using a family of quasi-convex functions. This waywould allow to find solutions to the optimisation problemwith a lower level of suboptimality with respect to thesolutions found for a pure non-linear optimisation problem.

7 AcknowledgmentsThis work has been supported by the CICYT WATMAN(ref. DPI2009-13744) of the Spanish Science andTechnology Ministry, the ‘Juan de la’ Cierva ResearchProgramme (ref. JCI-2008-2438), and the DGR ofGeneralitat de Catalunya (SAC group, ref. 2009/SGR/1491).

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