Agenda

1
Agenda

• Introduction to 7-SEAS
• Least Squares math and numerics
• Implementation in Aquis
• Examples
• Installation
• Summarize

2
7-SEAS

• State Estimation based Application System
• Advanced Leak Studies (ALS)
• Calibration Pipe wall roughnesses, diameters
• Valve state analysis
• Pump optimization
• ???

3
ALS

• Use deviation between measured (emulated)
  flows/pressures and simulated flows/pressures to
  estimate leaks. Aquis will insert leaks in the
  model to minimize the deviations between measured
  flows/pressures and simulated flows/pressures.
• ALS will not provide the exact position and size,
  but merely indicate a higher probability for
  leaks. The reason is lack of information (Not all
  pressures and flows are measured) and calibration
  errors and measurement errors will turn up as
  leak flows.

4
Pipe Calibration

• Pipe calibration is a correction of the
  frictional pressure drop
• ?P fcKQ2 ?Pelevation
• Based on the calibrated frictional pressuredrop,
  the corresponding diameter or roughness can be
  assessed.
• Note If the pressure drop is calibrated to be
  less than anticipated (fc lt 1 ), roughness may be
  calculated to a negative value and set to 0 as
  for a super-smooth pipe.

5
Advanced Leak Study

• Linear equation systems
• Overdetermined systems
• Least squares
• Non-linear systems
• Linear constraints
• Condition number
• Simple one-pipe example

6
Advanced Leak Study

• Linear equation system

• where
• A System matrix (size m x n)
• x Solution vector (length n)
• b Right hand side vector (length m)

7
Advanced Leak Study

• Square system m n

provided A is invertible (det(A) ? 0)
Overdetermined system m gt n The inverse of A
does not exist How do we determine x to best
fit the data?
8
Advanced Leak Study

• Different options
• Sum of absolute distance
• Min. of max. distance
• Least squares

9
Advanced Leak Study
Least squares for linear equation system
where the vector 2-norm
10
Advanced Leak Study
Normal equations
i.e.
provided exists (A has full rank)
The Moore-Penrose pseudo-inverse of a matrix A
provided A has full rank
11
Advanced Leak Study
Non-linear least squares
Newton-Raphson
where the Jacobi-matrix
Iteration k1
Find LS solution to Update solution vector
12
Advanced Leak Study
Linear constraints
Solve subject to
Approximate solution by reformulating
to unconstrained problem via weighting
with ? small
13
Advanced Leak Study
Matrix 2-norm
Condition number
Sensitivity of LS solution to perturbations in A
and b
14
Advanced Leak Study

• Simple one-pipe example

QN1
QN2
QP1
PN1
PN2
15
Advanced Leak Study

• Equation candidates
• Node mass continuity
• Pipe pressure drop
• Node pressure
• Node flow

16
Advanced Leak Study
All measurements available, 7 equations, 5
unknowns
17
Advanced Leak Study
Weighting to achieve fulfillment of continuity
and physics
with wg small (e.g. 0.01)
18
Advanced Leak Study
Weighting of individual measurements
where w1, w2, w3 and w4 are chosen to reflect the
confidence in the individual measurements.
19
Advanced Leak Study
Choice of weights
Pressure measurement weight relative to setpoint,
e.g.
Flow measurement weight relative to setpoint
and total flow, e.g.
where
20

• 7Flow is an Automation Server

21

• Guideline to conducting Advanced Leak Studies
• Gather all sorts of information
• Calibrate model
• Assess leak flow (NLM)
• Remove leak flows from nodes (Proportionally)
• Apply pipe leak potential factors
• Apply weights
• Run
• Test stability of solution by re-running on
  alternative data

22
Quick overview of 7-seas related plots

• Network plots
• Pipe leak potential
• Estimated pipe leak intensity
• Surface plots
• Node flow deviations
• Node pressure deviations
• Relative error
• Node leak potential

23
Pipe Leak Intensity