NAWMS: Nonintrusive Autonomous Water MonitoringSystem

Younghun Kim, Thomas Schmid, Zainul M. Charbiwala,Jonathan Friedman, Mani B. Srivastava

Networked and Embedded Systems Lab.Electrical Engineering Department

University of California, Los Angeles, CA 90095, USA{kimyh,thomas.schmid,zainul,jf,mbs}@ucla.edu

ABSTRACTWater is nature’s most precious resource and growing de-mand is pushing fresh water supplies to the brink of non-renewability. New technological and social initiatives thatenhance conservation and reduce waste are needed. Provid-ing consumers with fine-grained real-time information hasyielded benefits in conservation of power and gasoline. Ex-tending this philosophy to water conservation, we introducea novel water monitoring system, NAWMS, that similarlyempowers users.

The goal of our work is to furnish users with an easy-to-install self-calibrating system that provides information onwhen, where, and how much water they are using. The sys-tem uses wireless vibration sensors attached to pipes and,thus, neither plumbing nor special expertise is necessaryfor its installation. By implementing a non-intrusive, au-tonomous, and adaptive system using commodity hardware,we believe it is cost-effective and widely deployable.

NAWMS makes use of the existing household water flowmeter, which is considered accurate, but lacks spatial granu-larity, and adds vibration sensors on individual water pipesto estimate the water flow to each individual outlet. Com-pensating for manufacturing, installation, and material vari-abilities requires calibration of these low cost sensors toachieve a reasonable level of accuracy. We have devisedan adaptive auto-calibration procedure, which attempts tosolve a two phase linear programming and mixed linear ge-ometric programming problem.

We show through experiments on a three pipe testbedthat such a system is indeed feasible and adapts well tominimize error in the water usage estimate. We report anaccuracy, over likely domestic flow-rate scenarios, with long-term stability and a mean absolute error of 7%.

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.SenSys’08, November 5–7, 2008, Raleigh, North Carolina, USA.Copyright 2008 ACM 978-1-59593-990-6/08/11 ...$5.00.

Categories and Subject DescriptorsH.4 [Information Systems Applications]: Miscellaneous

General TermsAlgorithms, Design, Experimentation, Human Factors, Mea-surement, Performance, Theory

KeywordsAdaptive Sensor Calibration, Machine Learning, Water FlowRate Estimation, Nonintrusive and Spatially DistributedSensing, Tiered Information Architecture, Parameter Esti-mation via Optimization

1. INTRODUCTIONBob hears about water conservation all the time. He is

told that if every citizen consumed water moderately, oursupplies would last forever. But, what is moderate? Ishe wasting water? He decides to consult his utility bill,but quickly realizes that the monthly usage is for the wholehouse. What he needs is a system that can tell him notjust how much water is being used, but where it gets used,when, and by whom. Having a system prompt him in real-time about usage for every outlet in the house puts him ina better position to discover wastage and make corrections.

The problem Bob faces applies to any resource a house-hold consumes. Most utilities do not provide the spatial andtemporal granularity needed for household members to ef-fectively reduce their resource consumption, since it wouldmean higher installation and infrastructure cost. The sys-tems currently in place are adequate for their billing pur-poses.

McMakin [28], Ester [18] and Stern [38] show in theirstudies that providing fine-grained consumption informationhelps consumers pinpoint waste, and enables improved ef-ficiency. They also show that consumers are motivated toconserve, “do the right thing”, and reduce expenses. At amacroscopic level, conservation improves a country’s overallresource management. For example, the U.S. Environmen-tal Protection Agency claims that 3 trillion gallons of wa-ter could be saved each year if every household in the U.S.decreased its water consumption by 30 percent [41]. Thisresults in a dollar-volume saving of $49.3 million per day, ormore than $18 billion a year ! Water conservation is of evengreater financial significance as increasing purification cost

is being compounded by rapid demand growth. Americanpublic water supply and treatment facilities consume about56 billion kWh per year – enough electricity to power over5 million homes [41]. To make things worse, higher globaltemperatures are affecting our fresh water reserves in ice andsnow caps. The Water Poverty Index [10] illustrates that anurgent global initiative toward conservation is necessary.

The first step toward conservation is measurement [18,28,38]. For electricity, small power monitoring sensors haveemerged [15], but they are too expensive to be installed onevery electric appliance. For water, there is nothing anal-ogous available. Current consumer-grade water flow me-ters are meant to be installed inline, requiring non-trivialplumbing, best handled by a professional. Thus, conven-tional means of deriving high resolution data would comewith high installation cost. In some cases, where spatiallydense water monitoring systems are essential for correct op-eration, this is justifiable (e.g. irrigation systems [9] or phar-maceutical manufacturing plants).

The challenge is designing a system that provides the sameresolution at a cost and effort that is reasonable for house-hold use. To this end, we propose NAWMS: a scalable watermonitoring system capable of estimating water flow rate us-ing wireless sensor network technology. NAWMS uses inex-pensive vibration sensors attached externally to pipes. Thisreduces both cost and effort of installation. Inexpensive sen-sors and their un-controlled installation, however, introducenoise and variability in the measurements. NAWMS uses anovel feedback optimization formulation that continuouslyrecalibrates the sensors, thus minimizing the estimation er-ror for the total water consumption.

The contribution of this paper is three fold:

Well Defined Tiered Information Architecture:We introduce a well defined tiered information archi-tecture. The first tier is accurate but spatially coarsegrained. The second tier is less accurate but spatiallyfine grained. We exploit the two-tiered informationarchitecture to achieve spatially fine grained and ac-curate results(Fig. 1).

Realistic Modeling of a Water Monitoring System:

A sensor system needs to capture realistic constraintssuch as external noise, physical limitations and char-acteristics, and performance requirements. Our modeltakes these into account in order to compensate forside-effects.

Autonomous and Adaptive Calibration:A sensor system often suffers from complex calibration,and its periodic recalibration increases maintenancecost. A well defined performance metric that evaluatesthe system on the fly, and an optimization problemformulation that yields calibration parameters, enablethe system to autonomously and adaptively calibratethe water monitoring system while regulating its per-formance.

The rest of the paper is structured as follows. Section 2presents related work. We continue by describing the con-ceptual system architecture of NAWMS in Section 3. InSection 4 we formulate the mathematical challenges and in-troduce several optimization problems to solve them. Our

system evaluation can be found in Section 6, where we de-scribe our prototype and present the achieved system perfor-mance. Finally, Section 7 contains future work for NAWMS.

2. RELATED WORKThere are two broad classes of related work: (a) Infras-

tructure monitoring for activity classification and mainte-nance, and (b) water flow rate measurement techniques.

Infrastructure provides useful behavioral information aboutits inhabitants and often times this information can be ex-tracted through simple interfaces and means. For example,Patel [31] monitors the electrical noise within the power-lines of a house. They exploit the fact that each applianceintroduces a unique noise signature. By detecting and iden-tifying this signature, they can infer if an appliance is onor off but not its actual power consumption. Fogarty [21]investigated monitoring of the plumbing system by usingmicrophones on pipes to infer water activity in a household.Both systems are easy to install but employ complex cali-bration mechanisms in order to learn the detection patterns.Though these systems capture user behavior, we considerthem incomplete for conservation because they do not esti-mate actual consumption numbers.

At a larger scale, cities are struggling with the mainte-nance of aging water distribution and treatment systems.Stoianov [39] describes an interesting prototype sensor net-work that can monitor different water characteristics in real-time. In a lab setup, they demonstrated how one can detecta water leak in a pipe, by analyzing the frequency spectrumof an accelerometer. Even though they use similar hard-ware, Stoianov’s system has a completely different goal anduses a different mathematical approach.

Methods to measure the water flow rate, or more gener-ally the flow of liquid in pipes, is of great interest to manyfields. For example, chemical processes often require precisecontrol over fluid flow [37], agricultural irrigation networksneed monitoring to avoid over-watering [36], and utility com-panies deploy inline water flow meters for billing purposes.The methods could be divided into two categories: (a) openchannel water monitoring/discharged water monitoring and(b) water flow rate estimation in closed pipes.

Large scale irrigation systems generally use open channelsto distribute water to different areas. Trimmer [40] describesa way to estimate the discharged water flow rate by observ-ing the water level and pipe diameter. Bankston [3] provideslook-up tables that map manually obtained observations tothe flow rate of an open channel. Unfortunately, this methodis laborious and lacks real-time response.

Modern irrigation networks leverage wireless sensor net-work technology to monitor the water flow rate in each waterchannel. Additionally, they use this information to regu-late the flow rate in real-time. These distributed irrigationcontrol systems are a showcase for wireless sensors and ac-tuators that deliver optimal water volume for agriculturalirrigation [9].

Measuring the flow rate of a liquid in a closed pipe is fur-ther divided into two categories: inline direct flow measure-ment and non-intrusive flow estimation. The most commonexample of inline flow measurement is the main water flowmeter in a household that utility companies install. Thesemeters use a mechanical turbine that spins at an angularvelocity proportional to water flow rate. The constant ofproportionality is the exact diameter of the flow chamber,

Figure 1: NAWMS system and information architecture

which is estimated though factory calibration. Thus, count-ing the number of rotations of the turbine yields directly theflow rate in the pipe. The disadvantage of this technique isthat it needs to be installed between pipes segments, re-quiring plumbing expertise. This is feasible during initialconstruction, but retrofitting pipes is tedious and expensive.

Several techniques for a non-intrusive flow rate estima-tion exist. The most common one uses ultrasound [19].This technique is based on an ultrasound transmitter andreceiver pair that either measures the induced doppler shiftin the received signal, or the transmission time within theliquid medium. Unfortunately, commercially available de-vices [16] cost over $1000 a unit and require delicate instal-lation. Thus, they are reserved for industrial, or diagnostictesting purposes.

An innovative technique described by Evans [20] exploitsthe vibration induced by the flow of liquid in pipes. Thistechnique is potentially cost-effective, since it uses an ac-celerometer, and the data processing is trivial. However, itrequires careful calibration because the vibrations dependheavily on the pipe material used and the sensor-to-pipe at-tachment.

NAWMS combines the pre-installed inline flow meter (firstinformation tier) with non-intrusive inexpensive vibrationbased sensors (second information tier) in order to providean accurate per pipe flow rate. NAWMS exploits the highaccuracy of the first information tier to autonomously cali-brate the less precise, though spatially finer grained, secondinformation tier.

3. NAWMS SYSTEM ARCHITECTUREA sensor usually measures a physical phenomenon indi-

rectly. For example, a temperature sensor exploits thermalvariation of resistance, a light sensor measures photo con-ductivity, etc. Due to this indirectness, a sensor needs cali-bration to establish a mathematical mapping between sensorreadings and the physical phenomena. Some sensors can bebought factory calibrated, and some don’t even need cali-bration. One example is the inline water flow meter. Sinceits package dimensions are known during fabrication, thereis a well defined relation between turbine speed and flowrate. Unfortunately, it is infeasible to retrofit a householdor an entire building with these sensors. For this reason, weneed a system that indirectly measures water flow within apipe.

3.1 Tiered Information ArchitectureWe propose NAWMS, a two tiered information architec-

ture that consists of one accurate pre-calibrated sensor thatprovides us with the sum of all flows and an uncalibratedvibration sensor for each individual pipe (Fig. 1). This ar-chitecture has three characteristics:

• The first tier is reliable, and the service provider main-tains it. A user does not have to worry about its ac-curacy or correct operation.

• The calibration of the non-intrusive vibration basedwater flow sensors can be automated considering thecorrelation between the first and second tier informa-tion sources.

• The first tier is ’ground truth’ that is always available.This allows the system to compute and adapt the cal-ibration parameters continuously.

3.2 Hardware and Software ArchitectureThe system consists of three main hardware and software

components with the following associated challenges:

• Tapping Into Existing Infrastructure: The main watermeter provides accurate water flow rate for the entirehousehold. However, a mechanism is needed to extractreadings with high fidelity for the NAWMS system.

• Spatially Distributed Vibration Based Water Flow RateSensors: Accelerometers measure the vibrations ac-cording to the microscopic water flow model in a pipe.These accelerometers need to be calibrated in order toachieve a good estimate.

• Information Fusion Algorithm: Calibration requires alot of time and human intervention. In this paper wepropose several optimization formulations that enablethe system to calibrate its parameters and coefficientsautomatically using an open source optimization tool-box.

3.2.1 Tapping Into Existing InfrastructureThere are several commercially available products [1, 2,

25] that allow one to interrogate the main water meter inreal-time. Various types of “pulsers” [25] provide a pulsetrain that is proportional to the water flow rate in the pipe.A microcontroller based circuit, such as a “sensor mote”utilizing its built-in interrupts and timers, can easily decodesuch a signal providing real-time water flow information.

Another approach was described by Chueng [12]. Histechnique uses a hall effect sensor that picks up a varyingmagnetic field produced by the spinning collector unit. Hereports volume measurements with a resolution of 0.005 gal-lon.

Fortunately, a better solution is expected in the near fu-ture. The Automatic Meter Reading Association [2] andthe American Water Works Association [1] are developinga system that relays real-time meter information wirelesslyfor billing purposes. Two products, one from Germany [17],and another from the UK [35] are already available. Follow-ing the trend of utility companies simplifying their billingsystems through wireless technologies, we believe that ex-tracting data from the main water meter of a household inreal-time will soon be commonplace.

Figure 2: Microscopic view of the water flow in a pipe.

3.2.2 Non-intrusive Water Flow Rate MeasurementUsing Vibration Sensors

As Evans described in [20], the flow rate in a pipe is pro-portional to the vibration of a pipe. We describe this ef-fect in further detail in Section 4.1. A sensor network nodeequipped with a one dimensional accelerometer is adequateto capture this signal since variance in the measured accel-eration is a measure of vibration. Estimating variance iscomputationally simple, and thus can be done locally. Thisresults in low communication load. We describe other de-tails of the prototype used in Section 6.

3.2.3 Information Fusion, Optimization Toolbox, andAggregation Unit

The data collected by the vibration sensors is sent to thefusion center, where the samples and the reading from themain meter are “fused” to solve the mathematical optimiza-tion problems introduced in Section 4.2. The solution calcu-lates the calibration parameters for the individual vibrationsensors, which can then be used to map to real-time flow ineach pipe.

Many ready-to-use optimization problem solvers exist. Inour implementation, we use the CVX toolbox [8,14], an opensource convex optimization tool. Other advanced tools, suchas MOSEK [29], could also be used.

4. PROBLEM DESCRIPTION AND FOR-MULATION

This section is further decomposed into the underlyingpipe dynamic theories, their application to household plumb-ing, and the resulting formal optimization problem state-ments.

4.1 Water Flow Rate Estimation using Vibra-tion Sensors

4.1.1 Theory of OperationTo understand how flow rate and vibration in a pipe are

related, we cover the basics of its micro-model. Water mole-cules on average all travel in the main direction of flow,as depicted in Figure 2. However, many molecules collideagainst the pipe wall. According to the first law of thermo-dynamics, some part of this kinetic energy converts to heatas the turbulent eddies dissipate, but most of it translatesinto potential energy in the form of pressure [32]. The pipe,in turn, deforms converting potential to kinetic (during de-formation) and back to potential as deformation completes.The elasticity of the pipe material applies a restoring force.Evans shows [20] that vibration in a pipe results from thisenergy conversion cycle and is proportional to the averageflow rate within the pipe.

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35Vibration to Water Flow Rate

Vibration

Wat

er F

low

Rate

(L/s

)

Figure 3: Measured pipe vibration for different flow ratesfor a 3/4“ copper pipe.

The main result of Evan’s work is based on the fact thatthe velocity of a fluid at a point in the pipe can be decom-posed into the time average velocity(u, v) and fluctuatingvelocity(u′, v′). Although there is no net flow in the perpen-dicular direction to the pipe axis, the time averaged velocityin that direction v is zero, the time average of their productu′ · v′ are, in general, negative.

From a non-trivial derivation by Evan et. al., we see

∂2y

∂t2= −CEI ∂

4y

∂x4= −Cp′(x) (1)

where,C = g

AγA : cross sectional area of the beamγ : specific weight of the beamg : gravityEI : flexural rigidity.This essentially indicates that the transverse acceleration

of the pipe is proportional to the pressure fluctuations in thefluid.

The principle of operation is based on the relationshipbetween the standard deviation of the pipe vibration and themean flow rate of the fluid in the pipe. Blake stated in [5]that the generation of vibration by fluid motion involves thereaction of fluids and solids to stresses imposed by time-varying flow. For dynamically similar flows, the ratio of theflow fluctuations to the average flow is constant. Bird shows[4] this relationship by noting that the oscillatory term is thetime average of the absolute magnitude of the oscillation,given by

√m where m = u′2. They define this as “intensity

of turbulence”, which is a measure of the magnitude of the

turbulent disturbance, and is given by√mu

.From the definition of turbulent flow, the intensity of tur-

bulence expression is rearranged as

m

u2=

1N

PNi=1 [ui(t)− u]2

u2(2)

where,u: average velocityu: instantaneous velocity.

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25Vibration to Water Flow Rate

Vibration

Wat

er F

low

Rate

(L/s

)

Figure 4: Measured pipe vibration for different flow ratesfor a 3/4“ PVC pipe.

Multiplying both side by the number of points N and u2,and dividing by N − 1, we get

1

N − 1

NXi=1

[ui(t)− u]2 =NC

N − 1u2 = Ku2 (3)

This shows that the flow fluctuations are proportional tothe pressure fluctuations and the pressure fluctuations areproportional to the pipe vibration. It follows that the stan-dard deviation of the pipe vibration is proportional to theaverage flow rate. This result does not necessarily implythat the water flow rate in a pipe is linearly proportionalto the vibration of the pipe. Instead, it implies that it hasa non-linear but proportional relation due to the non-linearcharacteristics of vibration sensors, pipe structure, turbu-lence, etc.

The end result eq. 3 gives an important point that regard-less of the pipe mounting methods, shape, and topology, atany exposed point on a pipe we can find a signal that isstrongly correlated with the water flow rate at that point.

4.1.2 Vibration to Flow Rate ModelTo verify the theory explained in Section 4.1.1 we con-

ducted a simple experiment. We attached accelerometerson two different pipes, a copper and a PVC pipe. We mea-sured the vibration occurring on the pipes while changingthe flow rate of the water running through it. Figure 3 and4 illustrate the result. To find a mathematical relationshipbetween vibration and flow rate, we tested the applicabilityof various models, details of which we cover in Section 6.1.1.We found that a third order root function,

f(t) = α 3pv(t) + β

pv(t) + γv(t) + δ, (4)

where f(t) is the water flow rate, and v(t) the measuredvibration, fit the measured data well. We use this functionhenceforth to map the measured vibration to the actual wa-ter flow rate in the pipe.

4.2 Simple Pipe StructureIn this section, we formulate the large scale water flow

rate monitoring system for the pipe topology depicted inFigure 5. The system consists of the main water meter,

Figure 5: Simple pipe topology where one main pipe withflow rate M(t) splits up into N subpipes.

and one vibration sensor on each of the N sub-pipes. Thewater meter provides the system with the real-time waterflow M(t), an accurate measurement for the main pipe. Wedenote the flow rate in each sub-pipe i, fi(t). Since all thesub-pipes are connected to the main pipe (assuming no leaksin the system), we have

M(t) =

NXi=1

fi(t). (5)

where, fi(t) = αi 3pvi(t)+βi

pvi(t)+γivi(t)+δi as defined

in the last section.The goal of the calibration is to find the calibration pa-

rameters αi, βi, γi and δi for each sub-pipe. One possibilitywould be to attach a flow meter to each sub-pipe and cali-brate each pipe individually. But this is tedious and involvesmanual effort. A better way is formulating a mathematicaloptimization problem that estimates these parameters si-multaneously.

Assume that the sensors are synchronized, and that theysample the vibration every ∆t seconds. Thus, Equation 5in discrete-time becomes M(k∆t) =

PNi=1 fi(k∆t). After

collecting K samples of M(t) every ∆t second, we get Kequalities

M(k∆t) =

NXi=1

fi(k∆t) for k = 1, 2, 3, ...,K. (6)

We define

Mdef= [M(∆t),M(2∆t), ...,M(K∆t)]T

Fdef=

"NXi=1

fi(∆t),

NXi=1

fi(2∆t), ...,

NXi=1

fi(K∆t)

#T.

We now formulate Equation 6 as an optimization problembecause M = F holds true, unless there is water leakage.Thus, the optimization problem is written as

min ||M− F||1subject to 0 ≤ fi(k∆t) ≤ fi,max. (7)

Note that M(k∆t) and vi(k∆t) are measurements fromthe sensors, and αi, βi, γi and δi are the decision variables.Additionally, the problem consists of linear constraints only,and its objective function is also be expressed as a linearfunction [6]. Therefore, any available Linear Programmingsolver can solve this problem very efficiently, and it guaran-tees a global optimum.

Figure 6: More realistic pipe topology where sub pipes cansplit up into sub-sub pipes.

The solution to the problem yields the calibration param-eters for each individual pipe. Thus, no human interventionis necessary in order to calibrate the system, since every-thing is be automated.

4.3 General Pipe StructureThe plumbing system in a typical household is far more

complicated than the simplified structure assumed in theprevious section. We show in this section how even a com-plex structure can be reduced into several optimization prob-lems, and that each of these problems is similar to Equation7. This allows us to focus on the simple structure thereafterwithout loss of generality.

Consider the pipe structure depicted in Figure 6. Assum-ing no leakage, the total flow rate M(t) is equal to the sumof the flow rates of each sub pipe. Similarly, the flow rate ofeach sub pipe is to the sum of the flow rate of its sub-subpipes. Thus, from the law of mass conservation, we get thefollowing equalities:

fi(t) =

JiXj=1

fi,j(t)

M(t) =

NXi=1

fi(t), (8)

where, fi,j is the flow of sub-sub pipe j of sub pipe i andJi is the number of sub-sub pipes of sub pipe i. Similar

to fi(t), we define fi,j(t)def= αi,j 3

pvi,j(t) + βi,j

pvi,j(t) +

γi,jvi,j(t) + δi,j .With these equations, we formulate a newoptimization problem similar to Equation 7:

min ||M− F||1subject to 0 ≤ fi(k∆t) ≤ fi,max (9)

where

fi(k∆t) =

8<:PJij=1 fi,j(k∆t) if i-th sub pipe has

sub-sub pipesfi(k∆t) else

In general, the water flow rate in a pipe is the sum of thewater flow rate in its sub pipes. Therefore, any tree like pipestructure can be formulated in a form similar to Equation9.

4.4 Accounting for Vibration PropagationWe have assumed thus far that the vibration of a pipe

does not affect other pipes. However, since pipes are physi-cally coupled, vibration propagates to other pipe segments.This phenomenon needs to be accounted for in our equa-tions. Vibration propagation depends on pipe topology,pipe material, interconnects, and flow rate. Nevertheless,the induced vibrations are relatively small. As a first orderapproximation, we assume that propagated vibration is lin-early proportional to inherent vibration. This assumption isreinforced by actual experiments on our testbed, which aredescribed in Section 6.1.1.

Consider vibration only in pipe j. We define the vibrationpropagation from pipe j to pipe i as:

vi(t) = pi,jvj(t) (10)

where,vi(t) : Measured vibration on pipe ipi,j : Vibration propagation constant from pipe j to pipe

ivj(t) : Water flow induced vibration on pipe jGeneralizing this to the case where the measured vibration

on a pipe i is the superposition of all possible vibrations, weget

vi(t) =

NXj=1

pi,jvj(t)

pi,j = 1 if i = j0 ≤ pi,j < 1 else

We now update our optimization problem to include thevibration propagation coefficients:

min ||M− F||1subject to 0 ≤ fi(k∆t) ≤ fi,max

vi(k∆t) =

NXj=1

pi,jvj(k∆t) (11)

Introducing propagation coefficients adds a posynomialequality constraint to Equation 7. This renders the opti-mization problem non-linear and non-convex and harder tosolve. In the following sections, we explore two ways ofreformulating the problem as (a) a generalized geometricprogramming (GP) problem, by relaxing and restricting theconstraints with reasonably tight bounds, while conservingphysically meaningful values, and (b) a two phase linearprogramming (LP) problem by decomposing the optimiza-tion problem in two parts. It is well known that both, GP,and two phase LP problems can be solved in polynomialtime [7, 8]. Either of the techniques can estimate the cali-bration parameters. Depending on system performance re-quirements, an algorithm could potentially select which oneto use.

4.5 Mixed Linear Geometric ProgrammingModel

The posynomial equality constraints due to vibration prop-agation coefficients prevent us from solving the optimizationproblem as a GP problem [7,8]. Additionally, the objective

function is not in standard GP form. We require to re-formulate it since this class of problems, called SignomialPrograms, are in general NP-hard [13].

First, we introduce a slack vector sdef= [s1, ..., sK ]T ∈ RK+ .

We then reformulate Equation 11 using the slack vector tothe equivalent problem:

min

KXk=1

sk

subject to −s ≤ F−M ≤ s (12)

0 ≤ fi(k∆t) ≤ fi,max

vi(k∆t) =

NXj=1

pi,jvj(k∆t) (13)

By restricting the first constraint to be positive, we canrewrite equation 13 as

min

KXk=1

sk

subject to 0 ≤ F−M ≤ s

0 ≤ fi(k∆t) ≤ fi,max

vi(k∆t) =

NXj=1

pi,jvj(k∆t) (14)

This restriction shrinks the search space considerably, butcould make the problem infeasible. In most of the cases,however, the problem is still solvable [13]. According to [13],relaxing the posynomial equality constraints with reasonabletightness allows us to reformulate the problem in standardGP form.

We assume the following relaxation, since measured vibra-tion on a pipe is the sum of all propagated vibrations plusthe vibration induced by the water flowing through thatpipe. Thus,

NXj=1

pi,jvj(t) ≤ vi(t) ∀i (15)

As a consequence, we can rewrite Equation 14 as

min

KXk=1

sk

subject to F ≤ s + M

0 ≤ fi(k∆t) ≤ fi,maxNXj=1

pi,jvj(k∆t) ≤ vi(k∆t). (16)

According to [8] Section 7.3, this is a mixed linear GPproblem, as its objective function is linear and all constraintsare in a form either 1) (Posynomial) < (Affine) or 2) Posyn-omial inequalities.

Note that αi 3pvi(k∆t) + βi

pvi(k∆t) + γivi(k∆t) + δi is

posynomial because all the αi, βi, γi and δi are positive, andmeasured vibration is always positive. Therefore, since theclass of posynomial is closed under summation, all the con-straints on the left hand side are posynomial.

We can now solve Equation 16 very efficiently, roughlyproportional to max{n3, n2m,F}, where n is the number ofvariables, m the number of constraints, and F is the cost ofevaluating a posynomial’s first and second derivative. Ad-ditionally, we can guarantee a global optimum since it is amixed linear GP problem. Note that GPs are a special classof mathematical programs that can be converted to convexform using a change of variable. However, CVX [22] per-forms well on medium-scale GP problems, and it is thus notnecessary to perform this intermediate step.

Solving Equation 16 provides us with vibration propaga-tion coefficients and flow rate parameters, but its perfor-mance may be affected due to relaxed constraints. There-fore, we also introduce a two phase LP problem which cansolve the same set of equations.

4.6 Two Phase Linear Programming ModelWe reduce the complexity of solving Equation 11 by deriv-

ing it as an approximate LP problem. The propagation co-efficients are unknown and need to be estimated. However,since pipe topology is static and not expected to change sig-nificantly over time, we can estimate the propagation coeffi-cients opportunistically. After computing these coefficients,Equation 11 becomes a LP problem because the posyno-mial equality terms become linear equations. Therefore, theproblem can be decomposed into two problems: (a) vibra-tion propagation coefficient estimation, and (b) water flowrate parameter estimation.

4.6.1 Vibration Propagation Parameter EstimationIt is likely that only one pipe j has running water during

some period, and thus the system can measure the vibra-tion propagation between the pipes. In this case, vj = vj .The measured vibration on every other pipe i can then beestimated by

vi = pi,j vj . (17)

Additionally, the following holds true:

pi,j = 1 if i = j

pi,j =vivj

otherwise (18)

If the system behaved ideally, the measurements would beperfect, and the propagation coefficient computation wouldbe trivial. However, inherent measurement noise requiresus to reformulate this as a parameter estimation problem.We choose L1-norm minimization with linear inequality con-straints, since this tends to be less sensitive to significantoutliers compared to a least squares approach. According topreviously published results in sensor networks [11,23,24,27],inexpensive sensors on motes tend to have significant out-liers and faults.

The problem is solvable using a standard LP solver withthe appropriate linear inequality constraints. We define l asthe sample index, and let L be the total number of samples.The problem is then,

min˛˛

Vi − pi,jVj

˛˛1

subject topi,j = 1 if i = j0 ≤ pi,j < 1 else

(19)

where,Vi = [vi(1), vi(2), ..., vi(L)]T ∈ RL+Vj = [vj(1), vj(2), ..., vj(L)]T ∈ RL+

4.6.2 Vibration to Water Flow Rate Parameter Esti-mation

Once we compute the vibration propagation coefficients,the problem simplifies to a standard LP problem since thematrix P = [pi,j ] is now known and invertible. Finally, wesolve

min ||M− F||1subject to 0 ≤ fi(k∆t) ≤ fi,max

V(k∆t) = P−1V(k∆t) (20)

wherek = 1, 2, ...,KP = [pi,j ]V(k∆t) = [v1(k∆t), v2(k∆t), ..., vN (k∆t)]T ∈ RN+V(k∆t) = [v1(k∆t), v2(k∆t), ..., vN (k∆t)]T ∈ RN+

4.7 Correction Mechanism and PerformanceMetric

Solving the optimization problem above, the system canestimate the parameters required to map vibration measure-ments to flow rate. However, system characteristics canchange over time due to topological changes, seasonal tem-perature variations and aging. Therefore, we need a mech-anism that tests the system performance and recalibratesthe mapping, if necessary, through iteratively solving theoptimization problems.

We define a performance metric

Performancedef=

˛˛PNi=1 fi(t)−M(t)

M(t)

˛˛ (21)

where, fi(t) is the estimated flow rate in pipe i. If itsvalue is close to 0, we know the system is performing well,since the sum of the estimated flow rates has to be close tothe total flow rate M(t). However, if the metric exceeds aset threshold ε, the system needs recalibration.

A further improvement can be made based on this obser-vation. By normalizing each estimated flow rate fi with theratio of the total flow rate to the sum of all estimated flowrates, i.e.,

fi(t) =fi(t)M(t)PNi=1 fi(t)

(22)

we ensure thatPNi=1 fi = M(t) and is nearer the real flow

rate in the pipe.

5. SYSTEM DESCRIPTIONA user installs the vibration sensors on the pipes in their

plumbing and specifies the topology to formulate the opti-mization problem. Thereafter, the system runs autonomously.First, it collects vibration and water flow rate data to geta solution to the optimization problem. After the parame-ters are found, the system can estimate the individual flowrate in each pipe. Simultaneously, the system calculates andtests the performance metric introduced in Section 4.7. If

Figure 7: Algorithm flowchart for NAWMS. After an initialdata gathering, NAWMS solves an optimization problem toretrieve the calibration parameters. During system runtime,NAWMS monitors its own performance. If the performanceis bad, it recalibrates itself.

Table 1: Pipe Properties

Pipe Number Pipe Material DiameterPipe 1 Copper 3/4 in (19.05mm)Pipe 2 PVC 3/4 in (19.05mm)Pipe 3 PVC 1 in (25.4mm)

this metric exceeds a user specified threshold ε, the systemrecalibrates and solves the optimization problem again. Fig-ure 7 illustrates this algorithm in a flow chart.

6. EVALUATIONIn order to test and validate our system, we constructed

a plumbing testbed with three pipes of different materialsand diameters. Table 1 summarizes the pipe properties, andFigure 8 shows a picture of the setup. The main water pipe isequipped with a commercial water flow meter that generatesa pulse train proportional to the flow rate. We connectedthis pulse train to a Crossbow MicaZ mote that provides thefusion center with real-time flow rate measurements M(t).

Each branch of the testbed is equipped with an accelerom-eter. In our setup, we use a MicaZ mote with the MTS310sensor board. The MTS310 sensor board contains a 2DAnalog Device ADXL202 accelerometer that has a range of±2g. The node samples the axis perpendicular to the pipeat 100Hz and calculates the variance of acceleration every50 samples. This variance, vi(k∆t),is then sent to the fusioncenter for further processing.

In addition to the sensors, each branch has a ball typevalve at the end in order to control the flow rate within eachpipe. To get the per-pipe ground truth, we added anotherflow rate meter to the end of each pipe.

6.1 Evaluation of Water Flow Rate Model

6.1.1 Vibration to Water Flow Rate Model Valida-tion

Equation (3) suggests that the relation between the vi-bration variance and the flow rate is linear. Unfortunately,because of modeling inaccuracy and sensor characteristics,the relationship is non-linear as Figures 3 and 4 suggest. Us-ing a good fitting model is essential in order to minimize the

Table 2: Vibration to Water Flow Rate Fitting Model

Pipe Fitting Model Coefficient Fitting Error

Copper Pipef(t) = α 3

pv(t) + β

pv(t) + γv(t) + δ α = 0.1548, β = 0, γ = 0, δ = 0.0472 1.865

f(t) = βpv(t) + γv(t) + δ β = 0.1067, γ = 0, δ = 0.0923 2.030

f(t) = γv(t) + δ γ = 0.0478, δ = 0.1470 2.724

PVC Pipef(t) = α 3

pv(t) + β

pv(t) + γv(t) + δ α = 0.0896, β = 0, γ = 0, δ = 0.0160 1.058

f(t) = βpv(t) + γv(t) + δ β = 0.0530, γ = 0, δ = 0.0520 1.282

f(t) = γv(t) + δ γ = 0.0154, δ = 0.0884 1.892

PipesAccelerometers

Main Flow Meter

Figure 8: Image of the plumbing testbed showing the threepipes with accelerometers, and the main water flow meter.

estimation error. A commonly used fitting model would bea polynomial function. But our prior measurements showedthe vibration in a pipe saturates at some point. This effectis better captured with a square root, or even 3rd order rootcurve. Table 2 summarizes the different fitting curves, theirparameters, and the fitting error for two different types ofpipe material. We see that a 3rd order root curve has theleast fitting error for both pipes. Higher order root curvesmight fit better, but the increase in complexity (more cal-ibration parameters to estimate) does not ratify the smallgain in fitting error.

6.1.2 Vibration Propagation Among PipesTo verify the vibration propagation among the pipes, we

turned on one pipe, and measured the vibration occurringon all three pipes. Figure 9 depicts the case where pipe2 is running, and we plot the vibration of pipe 2 versusthe vibration of pipe 1 and 3. We see that our first orderapproximation in Equation 10 is reasonable and yields agood fit on the collected data, though the variation withinthe samples is very large. This might limit the achievableprecision in the flow estimation, but we illustrate in the nextsection how this can be ameliorated. Future research on thiscould further improve the propagation model by accountingfor vibration propagation time.

6.2 Flow Rate EstimationFor all of the following experiments, we used the two phase

LP problem to get the calibration information. In the firstexperiment, we investigate how well the system can track theflow in only one pipe. Thus, the flow rate in the main pipe

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1Vibration Propagation Parameter, pipe 2 to pipe 1

Vibration of Pipe 2

Vib

ratio

n of

Pip

e 1 measured

fit

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1Vibration Propagation Parameter, pipe 2 to pipe 3

Vibration of Pipe 2

Vib

ratio

n of

Pip

e 3 measured

fit

Figure 9: Vibration induced by water flow in pipe 2 prop-agates to pipe 1 and 3. NAWMS has to compensate for thiseffect to minimize estimation error.

0 20 40 60 80 100 120 140 160 1800.05

0.1

0.15

0.2

0.25

0.3Water Flow Rate Estimate and Actual Flow Rate in a Single Pipe

time (s)

flow

rat

e (L

/s)

TrueEstimate

0 20 40 60 80 100 120 140 160 180−0.05

0

0.05Estimation Error

time (s)

flow

rat

e (L

/s)

Figure 10: Water flow rate estimate for a single pipe.

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5Water Flow Rate Estimate of Pipe1 and Pipe2

time (s)

Flow

Rat

e (L

/s)

Estimated SumEstimated Pipe1Estimated Pipe2

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5True Water Flow Rate Estimate in the main pipe

time (s)

Flow

Rat

e (L

/s)

True SumTrue Pipe1True Pipe2

Figure 11: Water flow rate estimation of pipe 1 and 2. Thetop graph shows NAWMS estimated flow, the bottom is theground truth measured by the per pipe flow meters.

0 100 200 300 400 500 600−30

−20

−10

0

10

20

30Water Flow Rate Estimation Error of Pipe1 and Pipe2

time (s)

Flo

w R

ate

Est

imat

ion

Err

or (

%)

Sum ErrorPipe1 ErrorPipe2 Error

Pipe1 Error Mean : 4.24% STD : 9.68%Pipe2 Error Mean : −0.68% STD : 6.72%

Figure 12: Water flow rate estimation error for pipe 1 and2. The flow estimation is higher at low absolute flow rates(t = [0, 230] seconds) because of resolution problems in theADC.

was the same as the flow rate in the pipe under investigation.Figure 10 shows the result. On average, the estimation errorover the 180 second experiment was -0.49%, with a standarddeviation of 9.93 (In absolute numbers mean -0.0049L/s witha standard deviation of 0.014).

We now focus on the concurrent estimation of the flow ratein two pipes. Figure 11 compares the ground truth measure-ments (lower graph) to the estimated flow rate within thepipes and Figure 12 depicts its relative error. We see thatthe error of the sum of the two estimated flows is almost0. This is not surprising given that we normalize the twoindividual flows with the sum. Note that the relative er-ror reduces after about 230 seconds. The reason for this isthat the actual water flow rate almost doubled after thatpoint. While the accelerometer has enough sensitivity forlow vibrations, our current 10 bit ADC lacks the necessaryresolution to get below 0.004g, and thus a higher error atlower flow rates is to be expected. Our next prototype willtake care of this by providing a signal amplifier in these low

0 100 200 300 400 500 600

−5

0

5

Accumulated Water Usage Estimation Error in Pipe 1 and Pipe 2

time (s)

Wat

er V

olum

n (L

)

Sum ErrorPipe1 ErrorPipe2 Error

Figure 13: Accumulated water usage estimation error.The total water used was approximately 400L.

0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1

1.5

2

Water Flow Rate Estimate Error of Pipe1 and Pipe2

time (s)

Flo

w R

ate

(L/s

)

Pipe1Pipe2

0 50 100 150 200 250 300 350 4000

0.2

0.4

0.6

0.8

1Performance Metric Change Over Time

Performance

Recalibration Point

Figure 14: After an initial guess of the calibration param-eters, NAWMS recalibrates at t = 150 and the performancemetric improves considerably.

vibration regions. Figure 13 shows the error in the accumu-lated per pipe water usage and we see that it stays low, evenover a long period of time and high volume of water.

It is interesting to see how well our different modelingparameters influence the estimation accuracy. For this pur-pose, we ran the different optimization calculations fromSection 4 on the same experimental data. A summary ofthe result can be found in table 3. In the first calculation,we didn’t use any compensation (Figure 15a) and thus theestimation doesn’t account for vibration propagation, noruses the normalization correction. In the second calcula-tion, we accounted for vibration propagation only (Figure15b). We see that just accounting for this improves the esti-mation by more than a factor of 2. In the third calculation,we used the normalization correction described in Section4.7 (Figure 15c). The effect on the precision of the esti-mation is tremendous, since now the error of the sum ofthe two vibration estimates is forced to be 0. Last, usingall the correction mechanisms and accounting for vibrationpropagation and normalizing the two flows yields the bestresult (Figure 15d). This shows that our modeling of vibra-tion effects and correction mechanisms are efficient and helpreducing the overall estimation error.

Our last experiment shows the effectiveness of the per-formance metric. Figure 14 depicts the experiment result.First, we assumed some wrong calibration parameters and

0 100 200 300 400 500 600−0.2

−0.1

0

0.1

0.2Without Compensation

time (s)

Flo

w R

ate

Err

or(L

/s)

Sum ErrorPipe1 ErrorPipe2 Error

(a)

0 100 200 300 400 500 600−0.2

−0.1

0

0.1

0.2Only Account for vibration propagation

time (s)

Flo

w R

ate

Err

or(L

/s)

Sum ErrorPipe1 ErrorPipe2 Error

(b)

0 100 200 300 400 500 600−0.05

0

0.05Only with normalization

time (s)

Flo

w R

ate

Err

or(L

/s)

Sum ErrorPipe1 ErrorPipe2 Error

(c)

0 100 200 300 400 500 600−0.05

0

0.05With normalization + Account for vibration propagation

time (s)

Flo

w R

ate

Err

or(L

/s)

Sum ErrorPipe1 ErrorPipe2 Error

(d)

Figure 15: Adding the different water flow models and compensation techniques increases the precision of the water flowestimation. (a) is the baseline where no additional compensations are made. (b) accounts for vibration propagation. (c)normalizes the individual flow rates with the total flow rate in the main pipe, and (d) shows NAWMS’ performance whileaccounting for vibration propagation and normalizing the individual flow rates.

Table 3: Estimation improvement by using different flow rate models and compensation techniques

Scheme Pipe 1 Pipe 2Mean Error STD Mean Error STD

None -0.054L/s (-53.1%) 0.011 -0.094L/s (-63.0%) 0.017Vib Prop -0.016L/s (-15.8%) 0.010 -0.040L/s (-26.8%) 0.009

Normalization 0.016L/s (15.6%) 0.006 -0.013L/s (-8.5%) 0.006Both 0.010L/s (9.6%) 0.010 -0.007L/s (-4.4%) 0.011

used them to calculate the estimated flow rate. At t = 150seconds, the algorithm decides to recalibrate the system, us-ing the data collected between t = [0, 150]. The water flowrate estimation error after that point immediately drops toalmost 0, and the performance metric goes from almost 1down to approximately 0.4. This shows that using this per-formance metric, an efficient recalibration scheme can bedeveloped and used to keep the system accurate and effi-cient.

7. FUTURE WORKOur testbed shows that it is feasible to exploit the corre-

lation among vibration on each of pipes and the main meterreading to estimate water flow rate in individual pipe.

One limitation of the current system architecture is the re-quirement of having a sensor on each pipe. This can quicklybecome tedious and cost intensive as the pipes have to beexposed in order to install a sensor on them, and as thesize of the pipe topology increases. In a typical household,however, the number of water faucets is limited. For exam-ple, a bathroom usually has 5-7 water outlets, a kitchen has4 water pipes for a sink and a dishwasher, and a laundryroom has 3-4 water outlets. From a simple calculation, a 3-bed and 2-bathroom house has approximately 17-21 outletsexcluding outlets for the outdoor purpose. This basicallymeans that we need around 20 vibration sensors to estimatewater flow rate.

In a building, however, the number of pipes is much morethan this. To cope with this problem, we are currently ex-ploring ways of estimating the water flow rate with a fewer

amount of sensors. One possibility exploits the vibrationpropagation within the pipes. By detecting a specific sig-nature of propagated vibration, it might be possible to esti-mate the flow in pipes that do not have a sensor on them.

Since the vibration is mechanically propagating, pipes,physically mounted on a wall, are sensitive to external vibra-tion sources. Although our testbed showed that it is quiterobust to the external vibration sources such as foot steps,hitting the wall, and other typical activities in a room, areal house deployment can reveal interesting research ques-tions about how to cope with external noise sources to geta better estimate.

With a slight modification, we envision that our proposedframework for water monitoring could be applied to other re-source monitoring systems. For example, the electric energydistribution in a household has a very similar architectureto the water monitoring system. It has one main meter,and there exist simple to install sensors that need to be cal-ibrated in order to estimate the power consumption of anappliance [26]. The same is true for natural gas for heatingand cooking.

A further application of our system would be activity clas-sification, since water usage inherently requires human pres-ence, similar to [21]. However, the information that the sys-tem would provide could be more meaningful since it couldhelp people improve their natural resource efficiency.

Lastly, but not least, we want to investigate how a leakwill affect the overall system’s performance and how we candetect the leak. If a leak suddenly happens, one can imaginethat the vibration to water flow rate parameters change thus

the estimation error will become worse. The main questionis to find out how the parameters change or what othersignatures we need to have. Once this is done, the systemcould further alarm the owner if it detects that an unusualamount of water, or a continuous amount of water suggestinga leak or criminal activity. This can be especially helpful foroutdoor irrigation systems where the owner might not beable to immediately detect a water leak, because the visibleeffects disappear quickly.

8. CONCLUSIONWe have introduced a less intrusive, auto-calibrated, per

pipe water monitoring system that provides spatially finegrained real-time water usage previously not possible with-out extensive installation of inline sensors. In our work, weconsidered various pipe topologies and formulated severaloptimization problems that yield the system calibration co-efficients. The system makes extensive use of a two-tiredarchitecture and exploits pre-installed equipment that canusually be found in a household.

In our experimental results, we showed that vibrationbased water flow rate estimation is feasible and that theachieved individual per pipe flow rate error on average isless than 10%. By introducing a performance metric, wecan efficiently monitor the system calibration status, andautomatically decide if recalibration is necessary.

9. ACKNOWLEDGMENTSThis material is supported in part by the NSF under

award CNS-0520006, and by the Center for Embedded Net-worked Sensing at UCLA. Any opinions, findings and con-clusions or recommendations expressed in this material arethose of the authors and do not necessarily reflect the viewsof the listed funding agencies.

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