On-line Condition Assessment and Control of Water ...civil.iisc.ac.in/Murty_IITM_Aug08.pdfOn-line...

EWRE Division, Indian Institute of Technology Madras, Chennai- 36.

On-line Condition Assessment and Control of Water Distribution

and Gas Pipeline Networks

B. S. Murty Department of Civil Engineering

I.I.T. MADRAS


ACKNOWLEDGEMENT

Dr. H. Prashanth Reddy (Civil)Dr. S. Mohan Kumar (Chem.)Prof. Shankar Narasimhan (Chem.)

I.I.T. MADRAS


INTRODUCTION

Water distribution networks - pipes, tanks, reservoirs, pumps, and valves.

Water scarcity – need for increasing the efficiency and

reliability of supply in these networks.

– Leads to the studies on

• Monitoring

• Management and control


INTRODUCTIONINTRODUCTION

Natural gas to be transported from producing regions to consumption regions.

The biggest problem with the safe operation of the oil and natural gas pipelines is development of rupture leaks.

Delay in detecting leaks leads to loss of property and human life in fire hazards.


INTRODUCTIONINTRODUCTION

LDS can be classified as Software based automatic leak detection systems and field investigative leak detection systems.

SCADA based LDS is inexpensive and continuously monitors gas pipelines.


Hardware based techniques: Can detect small leaks,but expensive

Software based techniques: Cannot detect small leaks,but inexpensive

Inverse Transient methods: Computationally intensive,not suitable for on-line app.

Frequency analysis methods: Require suspension of normal operationsNot tested for complex networks

Most of the methods: Developed for WDN and not for gas pipelines

Problems with existing Methods


1

Node without demand

Delivery node

Sourcenode

2 3

4 5

6 7 8

[1] [2]

[3]

[4]

[5]

[6]

[7]

[8] [9]

Fig. 1: schematic of network


SIMULATION

Given

Pipe Characteristics

Source Pressure {function of time)

Demands {function of time}

Gas Composition {function of time}

Determine

Pressures & Flow rates

at all the points in the system


HOW DO WE DO THIS?

0)/( 2 =∂∂

+∂∂

tpcA

xM

01)2/(/)sin( 222 =∂∂

+++∂∂

tM

ApDAMMccpg

xp λθ

Continuity & Momentum Eqs.

For a Pipe Inclined at an Angle

Continuity Equation at a junction

Pressure Equality at a Junction


Solve the PDEs

Appropriate Numerical Method (FD / FV)

For Specified Boundary Conditions

(Specify Temporal

Pressure Variation at Source

& Demand Variation)


WHAT IS THE PROBLEM ?

FD Methods:

Time consuming !

Not Suitable for On-Line Applications(Leak Detection)


LEAK DETECTION

Assume a leak location & corresponding leak magnitude

Run the FD model for the above using pressure at source node and demand variation (Leak is treated as demand!)

Obtain Simulated pressures and flows at all the desired points where measurements are available


LEAK DETECTION (Contd.)

Determine the RMS error between measured & simulated pressures & flows

Solve the Optimization problem for leak magnitude & location such that the RMS error is minimized

This involves thousands of calls to the simulation code

Too much CPU time !!! (2 hours for a 1000 s run of one simulation, DT = 1 s)

Measurement noise is not factored into the methodology


STATE ESTIMATION

All we need for simulation: Pressure variation at source nodes and Demand variation at demand nodes

But we may have more measurements than this

All the above come with measurement noise


STATE ESTIMATION (Contd.)

Which measurements shall I consider in simulations ?

How do I know that the considered measurements are noise free?

State estimation reconciles all the measured data and gives out the expected (mean) state of the system which satisfies the governing equations

Absolutely important in leak detection via hypothesis testing


WHAT IS A TRANSFER FUNCTION MODEL?

Transfer functions relate M1 & P2 at any instant to M2 & P1 at that instant & Past values of M2 and P1

Using an approximation of Governing Equations

There is no need to discretize the pipeline as in FD methods

1 2

PipelineM1 & P1 M2 & P2


Linearize governing equations

Analytical solution in Laplace domain

Get transfer functions in Laplace domain

Take inverse Laplace in time domain


Down stream pressure equation in time domain

( )*

2 1 11

( )*

2 21

( )*2 2

21

( * ) * *(1 )* ( * )

* *(1 )* ( * )

* *(1 ) ( * )*

s s

s s

s s

T TN N iT T

s si

T TN N iT T

si

T TN N iT T

si

p N T k e e p i T

k e e M i T

k T e M i T eT

− − −

=

− − −

=

− − −

=

⎡ ⎤∆ = − ∆ +⎢ ⎥

⎣ ⎦⎡ ⎤− − ∆ −⎢ ⎥⎣ ⎦

⎡−− ∆

⎣

∑

∑

∑ 2 22* ( * )s

k T M N TT

⎡ ⎤⎤+ ∆⎢ ⎥⎢ ⎥

⎢ ⎥⎦⎣ ⎦

(11)


Upstream flow equation in time domain

( )*

1 21

( )*1

11

11

( * ) * (1 ) * ( * )

T - * (1 ) * ( * ) *T

T + * ( * )T

s s

s s

T TN N iT T

s si

T TN N iT T

si

s

M N T e e M i T

e p i T e

p N T

− − −

=

− − −

=

⎡ ⎤∆ = − ∆ +⎢ ⎥

⎣ ⎦⎡ ⎤⎡ ⎤

− ∆⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

∆

∑

∑

(12)


TskF PP +

=1

111,2 Ts

sTF PM +

=1

11,1 Ts

F MM +=

11

2,1 TssT

kF MP ++

−=1

)1( 222,2

Ψ=ek1

⎟⎠⎞

⎜⎝⎛ Ψ+= Ψ 22/

2 2411u

DALek λ

⎟⎠⎞

⎜⎝⎛ Ψ+Ψ−= Ψ 2

2

22/

241

611

2 cD

uLeT λ

⎟⎠⎞

⎜⎝⎛ Ψ+= Ψ 2

22/

1 2411

c

ALeT

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

Ψ+

+=

241

161

22

2

2cD

uL

uDT

λ

λ

22

sin2 c

gL

c

uu

DL θλ

−=Ψ

By expanding Transfer functions


Complete discrete model in the time domain for the entire network:

Combine above Eqs.(11) and (12) for all the pipe elements with

Continuity equation and pressure equilibrium equations at thejunctions

Continuity equation and pressure drop equation at valves

Continuity equation and equations describing compressor operation.


Formulation of the State Estimation Problem

Resulting system of equations for the entire network: Linear

Ax + Bu = 0 (13)

vector x: All measured variables (corresponding to time instants (N-n)T to NT,

vector u: All unmeasured variables (corresponding to time instants (N-n)T to NT).


The matrices A and B depend on the pipe parameters, sampling period, compressibility factor and friction factor.

Matrices A and B are time dependent


The best estimates of variables x and u, for a given set of measurements y for the variables x, can be obtained by solving the following weighted least squares estimation problem

Minimize

Subject to Ax + Bu = 0

Matrix Q is the covariance matrix of errors in measurement

Standard reconciliation problem

Crowe’s Projection Matrix technique

)()( 1 xyQxy T −− −


Obtain the state estimate for the current time instant NT in a recursivemanner by utilizing the estimates obtained for all the previous times

The constraint Eq. (13) is re-cast as given below.

where

Measured and unmeasured variables corresponding to the current time instant NT

and

are the corresponding sub-matrices of A and B.

Ax Bu c+ =, x u

,A B


Vector c : Weighted sum of the estimated flows and pressures at the previous time

Just enough variables are specified

Solution: objective function value = 0simulation problem

More measurements (specifications) than the minimum required to solve the problem are given:

Formulation gives a best fit solution

Takes into account the inaccuracies

state estimation


RESULTS AND DISCUSSION

• Second order MacCormack explicit FD method is used to validate the proposed TF model.

• Network (fig. 1) consists of 8 nodes and 9 pipe elements

• Slow transient caused by variation of demand at one of the demand node is simulated with Bench mark model (FD model) and proposed model


1

Node without demand

Delivery node

Sourcenode

2 3

4 5

6 7 8

[1] [2]

[3]

[4]

[5]

[6]

[7]

[8] [9]

Fig. 1: schematic of network


Demand Variation with with time at node 8

0.60.70.80.9

11.11.21.31.4

0 1000 2000 3000 4000 5000 6000

Time (sec)

Dem

and

(MM

SCM

D)

Fig. 2 Specified Demand variation at node 8


58.8

58.9

59

59.1

59.2

59.3

59.4

0 1000 2000 3000 4000 5000 6000

Time (sec)

Pres

sure

(kg/

cm2 )

Bench mark model Proposed model (dt = 1 sec) Proposed model (dt = 15 sec)

Comparison of pressure at node 8

RESULTS AND DISCUSSION RESULTS AND DISCUSSION –– DYNAMIC DYNAMIC SIMULATIONSIMULATION


Fig. 12: True and estimated mass flow rates at node 1: Case-2

1.175

1.275

1.375

1.475

1.575

1.675

1.775

0 1000 2000 3000 4000 5000 6000

Time (sec)

Mas

s flo

w ra

te (M

MSC

MD

)

Estimated mass flowrate without redundancy Estimated mass flowrate with redundancyTrue mass flow rate


Fig. 14: True and estimated demands at node 5: Case – 2

0.485

0.49

0.495

0.5

0.505

0.51

0.515

0 1000 2000 3000 4000 5000 6000

Time (sec)

Dem

and

(MM

SCM

D)

Estimated demand without redundancy (dark blue)Estimated demand with redundancy (pink)True demand (yellow)


0.014620.02116P8

0.016820.02365P7

0.017520.02444P6

0.0010.00313D5

0.017030.02393P5

0.0001230.001087F4

0.017090.02394P4

0.018370.02521P3

0.0001880.004563F2

0.018680.02480P2

0.0001060.05489F1

0.004920.01000P1

Redundant Measurements with noiseJust specified with noiseVariable

Table 5: Reduction in RMS error with increased redundancy


0.30.350.4

0.450.5

0.550.6

0.650.7

0.75

0 1000 2000 3000 4000 5000 6000

Time (sec)

Dem

and

(MM

SCM

D)

Fig. 17 Estimation of unknown demand at Node 5


CONCLUSIONS CONCLUSIONS –– STATE ESTIMATIONSTATE ESTIMATION

Test problems on the example network indicated that the proposed method is 25 times faster than the explicit finite-difference approach.

It was also demonstrated that the proposed approach can be used to estimate unknown demands.

The above features, coupled with the computational efficiency, make the approach ideally suited for on-line leak detection and identification.


LEAK DETECTION METHODOLOGYLEAK DETECTION METHODOLOGY

In order to detect a leak online, we will call state In order to detect a leak online, we will call state estimator at each sampling time instant.estimator at each sampling time instant.

ObjFunction > Threshold => possible leakObjFunction > Threshold => possible leak

We will hypothesize a leak in every branch of the We will hypothesize a leak in every branch of the pipeline network and determine best fit Dpipeline network and determine best fit DLL, X, XLL based on based on measurements [t, t+WT] in each branch. measurements [t, t+WT] in each branch.

The hypothesis that best fits the data among all the The hypothesis that best fits the data among all the hypotheses then determines the branch, location and hypotheses then determines the branch, location and magnitude of the leak magnitude of the leak



PP11,M,M11

11

PP22,M,M22

22

Branch iBranch i

xxii LL--xxii

bbll = unknown leak magnitude= unknown leak magnitude

xxiiLL--xxii

PP11,M,M11PP22,M,M22

Branch iBranch i11 Branch iBranch i22PPLL,M,MLL

Unknown demand DUnknown demand DLL

Leak pipe modelLeak pipe model


Leak detection hypothesis Leak detection hypothesis –– optimization problemoptimization problem

1

m ,uM in ( ) ( )

Ty m Q y m

Sub jected toA m B u c

−− −

+ =

The DThe DLL is unknown and it is part of u vector is unknown and it is part of u vector



hypothetical series pipelinehypothetical series pipeline

RESULTS AND DISCUSSION RESULTS AND DISCUSSION -- LEAK LEAK DETECTION USING SIMULATIONSDETECTION USING SIMULATIONS



Available instrumentation:Available instrumentation:

Pressure and flow measurements are available at both Pressure and flow measurements are available at both the ends.the ends.

Intermediate pressure measurements are available at SVIntermediate pressure measurements are available at SV--1, SV1, SV--4, SV4, SV--5 and SV5 and SV--7.7.

Total Total six pressure measurementssix pressure measurements and and two mass flow two mass flow measurementsmeasurements..

This is the basic instrumentation level considered for leak This is the basic instrumentation level considered for leak detection simulations but additional pressure detection simulations but additional pressure measurements added to improve the leak isolation measurements added to improve the leak isolation efficiency.efficiency.


Discretized form of hypothetical series pipelineDiscretized form of hypothetical series pipeline




Natural gas composition used isNatural gas composition used is:: CHCH44 93.42, N93.42, N22 0.12, CO0.12, CO22

2.36, C2.36, C22HH66 1.76, Propane 1.35, i1.76, Propane 1.35, i--Butane 0.31, nButane 0.31, n--Butane Butane 0.32, i0.32, i--Pentane 0.01, nPentane 0.01, n--Pentane 0.08, nPentane 0.08, n--Hexane 0.01.Hexane 0.01.Dynamic Viscosity of the natural gas was taken as Dynamic Viscosity of the natural gas was taken as 0.0000125 N s m0.0000125 N s m--22..Boundary conditions for the transient test:Boundary conditions for the transient test:P1 (Pressure at nodeP1 (Pressure at node--1) = 45.0 kg/cm1) = 45.0 kg/cm22;;Normal demand at consumption node is 60500 SCMH but Normal demand at consumption node is 60500 SCMH but is a function of time to create unsteady flow conditions;is a function of time to create unsteady flow conditions;



With noise + no filter + PT at every nodeCategory-6

With noise + no filter + additional ten PT’s to existing instrumentation

Category-5

With noise + filter (90% (90% weightageweightage to past to past data) data) + existing instrumentation

Category-4

With noise + filter (80% (80% weightageweightage to past to past data) data) + existing instrumentation

Category-3With noise + existing instrumentationCategory-2

Without noise + existing instrumentationCategory-1Test DescriptionS. No

CATEGORIES OF TESTS



Improvement in the leak detection results with extra nineteen Improvement in the leak detection results with extra nineteen pressure measurements added to the existing instrumentationpressure measurements added to the existing instrumentation

151.26123.760500.42195.08195.59

33102723.930250.94194.56195.58

12815.71020.712102.89192.61195.57

184.36310.160500.5103.56103.066

4914.83472.530250.77103.83103.065

18015.21393.612100.91102.15103.064

212.46193.860500.2453.9753.733

529.62735.330250.5253.2153.732

17531.91596.312100.8952.8453.731

(SCMH)(SCMH)(km)(km)

Delay in leak detection time (sec)

% error in estimated leak magnitude

Estimated leak magnitude

Magnitude of leak tested

Error in leak location

Estimated leak location

Leak location (km)

S. No


CONCLUSIONS CONCLUSIONS -- LEAK DETECTION LEAK DETECTION USING SIMULATIONSUSING SIMULATIONS

Proposed methodology is validated for 2%, 5% and 10% Proposed methodology is validated for 2%, 5% and 10% leaks using a series pipeline and a pipeline network.leaks using a series pipeline and a pipeline network.

Accuracy of the proposed method decreases when Accuracy of the proposed method decreases when measurement noise is present. measurement noise is present.

Results for a total of 66 numerical runs indicated that the Results for a total of 66 numerical runs indicated that the proposed methodology works very well if noise level in proposed methodology works very well if noise level in the measured data is low. In case of noisy data, the the measured data is low. In case of noisy data, the proposed method performs well if there is a sufficient proposed method performs well if there is a sufficient redundancy in the measurements.redundancy in the measurements.


RESULTS AND DISCUSSION RESULTS AND DISCUSSION -- LEAK LEAK DETECTION USING LAB EXPERIMENTSDETECTION USING LAB EXPERIMENTS

Experimental setup


RESULTS AND DISCUSSION RESULTS AND DISCUSSION -- LEAK LEAK DETECTION USING LAB EXPERIMENTSDETECTION USING LAB EXPERIMENTS

Network7.656.026.7124.68206.9200250

Network13.0212.0103.0499.68203.5200250

Network10.059.075.0674.68203.8200200

Network4.836.056.949.68227.2200150

Series7.386.030.9224.68204.4200250

Series12.1312.0100.4299.68201.5200250

Series9.729.075.0574.68203.6200200

Series6.346.045.9449.68206.3200150

EstimatedActualEstimatedActualEstimatedActual

Pipelineconfiguration

Leak Magnitude(SLPM)

Leak Location from inlet end (m)

Time of Leak (sec)Flow rate(SLPM)

Validation of proposed leak detection methodology using laboratory experimental data


CONCLUSIONS CONCLUSIONS –– LAB EXPERIMENTSLAB EXPERIMENTS

A total of 72 experiments were conducted by changing initial flow rate, leak location, leak magnitude, and network configuration (series and network).

In series pipeline, the error in the magnitude estimation was less than 10% in 65% of the cases and the error in the magnitude estimation was less than 15% in 78% of the cases. Maximum error in estimation (31%) occurred in one test.

The proposed method located the leak within 3 m (2.5% error based on the total length of the pipe) from its actual location of occurrence in most of the cases (32 out of 36 tests). (series pipeline)

The estimated leak location was within 3 m (2.5% error based on the total length of the pipe) from its actual location of occurrence in 17 out of 36 cases. (network)

The magnitude error in the estimation was less than 10% in 50% of the cases. The magnitude error in the estimation was less than 15% in 72% of the cases. (network)


The leak detection and identification method was implemented on-line on the LANCO pipeline, owned and operated by the GAIL (India) Ltd.

The pipeline is used to supply natural gas from Tatipaka to the LANCO power plant at Kondapalli.

The existing instrumentation consists of six pressure sensors one each at Tatipaka, Dindi (SV-1), Mortha (SV-4), Tadepalligudem (SV-5), Koppaka (SV-7) and Kondapalli, a gas chromotograph at Kondapalli, mass flow meters at Kondapalli and Tatipaka, and temperature sensors at Kondapalli and Tatipaka.

RESULTS AND DISCUSSION RESULTS AND DISCUSSION –– FIELD LEAK FIELD LEAK DETECTION TESTSDETECTION TESTS


Hypothetical Series pipeline systemHypothetical Series pipeline system

Pipeline characteristics:Pipeline characteristics:

LengthLength = = 204.7 km204.7 km

DiameterDiameter = = 0.443 m (ID)0.443 m (ID)

RoughnessRoughness = = 250 microns250 microns

Natural gasNatural gas is flowing through the systemis flowing through the system

Average temperatureAverage temperature = = 302 K302 K

Flow measurements sampling intervalFlow measurements sampling interval = = 10 seconds10 seconds




Schematic of LANCO pipeline



3 4.6717.51:58 PM1:51 PM6

6 125.1139.06:35 PM6:06 PM5

10 66.061.91:07 PM12:55 PM4

3 113.4139.07:31 PM7:07 PM3

10 162.1139.06:49 PM6:25 PM2

1.5 63.661.910:20 AM10.20 AM1

EstimatedActualEstimatedActual

Estimated Leak Magnitude

(% of total flow)

Leak Locationfrom Tatipaka (km)

Time of LeakField Test No.

Validation of the proposed leak detection methodology using field tests


Objectives

Overall objective - develop techniques for monitoring and control of water distribution networks.

The specific goals are to

– Monitor the health of the pipes by online estimation of pipe roughness coefficient

– Develop an online control strategy for optimal operation of water distribution network

– Validate the developed methods through simulation of large scale networks


State Estimation

State estimation – estimate flows, pressure, outflows from noisy measurements.

– Nonlinear constrained optimization problem

Objective

Constraints– Continuity (flow balance) equation at each node– Loop equations that relate the pressure drop variables – Correlation for energy losses due to friction (Hazen- Williams

correlation is used here)

( ) ( )YYYYMinT ˆ~ˆ~ 1 −∑−= −φ


MethodologyMethodology

Derive reduced optimization problem using graph theoretic concepts

Reduced number of constraints is equal to number of independent loops

Reduced optimization problem is then solved using Successive

Quadratic Programming technique

Gradient of objective function and reduced constraints are derived

analytically


Schematic diagram of Tiruppur network

Network Details

Pipes = 71

Nodes = 46

Source Node = 1

Demand node = 45


Case Studies – Tiruppur network

No

Yes

Yes

Yes

All 45 nodes

All 45 nodes

All 45 nodes

All 45 nodes

1

1

1

1, 35 to 46

-

-

1 to 12

1 to 12

1

2

3

4

Demand at nodes

Pressures at nodes

Flows in pipes

Noise(Yes/No)

Measured variables

Case

1001001005.0

9594942.0

8877771.0

6865480.5

2621150.1

Case-4Case-3Case-2% pipes (flow)% error

less than


Unknown Demand Estimation

0.20.002980.0029727

3.60.000520.0005119

6.30.001180.0012616

15.70.004320.0037337

5.20.00120.0012616

16.10.004330.003733613.40.004230.0037335

% ErrorEstimated demand(m3/s)

Actual demand,

(m3/s)

Demand unknown at

nodesCase


Parameter Estimation in WDN

Given measurements flows, pressures and outflows estimate pipe roughness coefficients

State estimation is extended to perform combined state and parameter estimation

Step-1: Reduce number of parameters by grouping pipes

– K-means clustering algorithm used to group pipes based on pipe diameter and age

Step-2: Estimate states and reduced set of pipe coefficients

– Graph theoretic reduction procedure extended


Case studies for parameter estimation

1201206-9, 24-28,34-55, 70, 71

9012010-23, 29-33, 57-69

601201-5, 56

Case - 11Cases- 8,9 and 10

Pipe No’s


Measurements for different cases

YesAll 45 nodes1, 23-321-2411

YesAll 45 nodes1, 23-321-2410

YesAll 45 nodes1, 23-321-159

YesAll 45 nodes1, 23-261-58

Demandat nodes

Pressuresat nodes

Flowsin pipes

Noise(Yes/No)

Measured variablesCase

For cases 8,9,10 and 11– noise added 1% of the true value


Comparison of actual and estimated HWC

4.59125.519.15109.0236.2676.481206-9,24-28,34-55,70,71

8.331304.03115.168.3313012010-23,29-33,57-69

4.3114.846.46112.246.79111.851201-5,56

Error %

Estimated HWC

Error %

Estimated HWC

Error %

Estimated HWC

Case-10Case-9Case-8Actual HWC

Pipe No’s

8.331301206-9, 24-28,34-55, 70, 71

18.72106.859010-23, 29-33, 57-69

7.3064.38601-5, 56

Error %Estimated HWC

Actual HWC

Pipe No’s

Case-11


Control of water distribution network

Control objective

Equitable distribution of water

Manipulated variables -

valve openings - continuous valves

Solution strategy – Model predictive control


Control problem formulation

0 < U < 1 for continuous valvesLoop constraintsMinimum pressure specifications

Constraints

Inferential Scheme

Objective

for P time periods

( )1

2

, ,... 1 1

ˆ ˆmind

k k M

N Psp pmmi k j i k j ku u i j

f d d d+ −

+ += =

= − −∑∑

ˆ ˆpmmk k kd d d= −

| 1 |ˆ ˆ ˆpm m

k k k k kd d d−= −


Control strategy – Flow chart

Modular implementation in C interfaced with FORTRAN optimizer

Control Algorithm Process(WDN)

StateEstimator

Valve settings

Measurements

Demand setpoint profile External disturbances

Disturbance estimated,States estimated

Extended Period Simulation

Valve settings(M-time period)

States(P-time period)


Sample network with only continuous valves

Continuous valves = 11 and 12Demand nodes = 5 and 11

Insufficient2

Sufficient1

Water available in

reservoir

Case

Case studies


Case- 1: Control algorithm results

Demand at nodes are met since sufficient water available

Demand at node - 5

00.00050.001

0.00150.002

0.00250.003

0.0035

0 2 4 6 8 10 12 14 16 18 20 22 24

Time, hr

Dem

and,

m3 /s

Demand Estimated outflow (M=1,P=1)

Demand at node - 11

00.0010.0020.0030.0040.0050.0060.007

0 2 4 6 8 10 12 14 16 18 20 22 24

Time, hrD

eman

d, m

3 /sDemand Estimated outflow (M=1,P=1)



Demand at nodes are not met since sufficient water is not available

Demand at node-5

0

0.002

0.004

0.006

0.008

0.01

0.012

0 2 4 6 8 10 12 14 16 18 20 22 24

Time, hr

Dem

and,

m3 /s

Demand Outflow (M=1,P=1) Outflow (M=1,P=5)

Demand at node-11

0

0.005

0.01

0.015

0.02

0.025

0 2 4 6 8 10 12 14 16 18 20 22 24

Time, hr

Dem

and,

m3 /s

Demand Outflow (M=1,P=1) Outflow (M=1,P=5)


Demand at node-5

00.00050.001

0.00150.002

0.00250.003

0.0035

0 2 4 6 8 10 12 14 16 18 20 22 24

Time, hr

Dem

and,

m3 /s

Demand With disturbance correctionWithout disturbance correction

Demand at node-11

0

0.001

0.002

0.0030.004

0.005

0.006

0.007

0 2 4 6 8 10 12 14 16 18 20 22 24

Time, hrD

eman

d, m

3 /sDemand With disturbance correctionWithout disturbance correction

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