Nonlinear estimation of a power law for the friction in a pipeline

Nonlinear estimation of a power law for the friction in a
pipeline
(IFAC MICNON 2018)
n m N
Authors: Lizeth Torres and Cristina Verde
ftorreso@iingen.unam.mx
http://lizeth-torres.info/
20th June 2018
Supported by Proyecto 280170, Convocatoria 2016-3, Fondo Sectorial CONACyT-Secretaría de Energía-Hidrocarburos.

Nonlinear estimation of a power law for the friction in a pipeline
Contents
1 Motivation
2 Contribution
3 Background
4 A power law for the total head losses
5 Identification approach
6 Design of the observers
7 Experimental results
Lizeth Torres | 20th June 2018 2 / 31

Nonlinear estimation of a power law for the friction in a pipeline | Motivation
Contents
1 Motivation
2 Contribution
3 Background

Motivation of the research
Contributing to the proper management of distribution pipelines.
A proper management requires some tasks such as ...
The diagnosis and prognosis of the pipeline components.
The control for the adequate distribution of the fluids.
The solution of optimization problems.
⇓
Algorithms based on models formulated from physical laws that govern the
flow behavior throughout the pipelines.

Disadvantage
Algorithms based on governing equations need to be frequently updated to
avoid being invalid because of the nonstop use and the passing of time.
Energy dissipation changes.
Affected Parameters
Roughness εa
.
Diameter φ.
a
It is not simple to measure it
Aging deterioration consequences
Corrosion
Tuberculation
Erosion
Mineral deposits

Motivation
Main goal of this article
Proposing an algorithm to continuously estimate the energy dissipation in
a pipeline without measuring ε, φ and L.
Why?
To avoid miscalculations due to changes of the physical parameters that
could affect the performance of model-based algorithms.

Nonlinear estimation of a power law for the friction in a pipeline | Contribution
Contents
1 Motivation
2 Contribution
3 Background

Nonlinear estimation of a power law for the friction in a pipeline | Contribution
Contribution
A power law to represent the energy dissipation mainly caused by
friction in a pipeline.
A simple method to estimate such a power law with the following
characteristics:
1 It is based on nonlinear state observers.
2 It only requires two pressure head recordings and a flow rate
measurement.
3 It performs the estimation in short-time.

Nonlinear estimation of a power law for the friction in a pipeline | Background
Contents
1 Motivation
2 Contribution
3 Background

The classic way to compute the energy dissipation
In pipelines, energy dissipation is called head loss and can be divided into
two main categories: major losses associated with energy loss per length of
pipe and minor losses associated with bends, fittings, valves, that act against
the fluid and reduce its energy.
∆Htot = ∆Hf + ∆Hm
∆Htot = f
L
φ
v2
2g
+
N
i=1
ki
v2
2g
= f
L
φ
+
N
i=1
ki
v2
2g
∆Htot =


f
L +
φ N
i=1 ki
f
φ



v2
2g
*Darcy-Weisbach equation

Total head loss in pipeline
If Leq = L +
φ N
i=1 ki
f then
∆Htot = f
Leq
φ
v2
2g
Leq is known as equivalent length, which actually is the virtual length of a pipeline
with devices as if it were a horizontal pipeline without devices.
Note: If a device (e.g. a valve) in the pipeline is added or changed, the equivalent
length will change and consequently the total head loss.
The computation of f depends on the flow regime.

Total head loss in pipeline
For laminar flow:
f = κ/Re
where κ depends on the pipeline geometry. For circular pipes κ = 64.
For turbulent flow (natural regime in pipelines): f can be accurately calculated by
using the Colebrook-White (CW) equation
1
f
= −2 log10
ε
3.71φ
+
2.51
Re f
Some disadvantages: ε is required, it is an implicit equation, it represents the
complete turbulent regimen of the Moody diagram! We don’t need that, we only
need to represent ∆Htot for the region where our pipelines work.
These disadvantages are reasons to propose new formulas to represent ∆Htot.

Another reasons to propose new formulas for ∆Htot
An identifiable equation
To have an equation with empirical parameters that are easy to estimate
A differentiable equation
Some optimization and identification algorithms require the derivative of
the friction.

Some proposed alternatives
1 Hazen William equation
∆Htot =
10.67
C1.852
LeqQ1.852
φ4.8704
2 Wood equation
∆Htot = (a(ε) + b(ε)Re−c(ε)
)
f
Q2Leq
2gφA2
r
3 Valiantzas equation
∆Htot = Leq
k0Q2
φ5.3
m
, k0 = 0.0126ε0.3
4 Quadratic (Prony) equation
∆Htot = aQ2
+ bQ
5 Rojas equations 1
∆Htot =
Leq
2gφA2
r
(θ1Q2
|Q| + θ2Q|Q|), ∆Htot =
Leq
2gφA2
r
θ1Q|Q| + θ2|Q|
1
Rojas Jorge et al. On-Line Head Loss Identification for Monitoring of Pipelines,
Safeprocess 2018, Poland.

Nonlinear estimation of a power law for the friction in a pipeline | A power law for the total head losses
Contents
1 Motivation
2 Contribution
3 Background

Nonlinear estimation of a power law for the friction in a pipeline | A power law for the total head losses
Our proposition
∆Htot = ΩQ1+γ, where Ω =
α
β
.
α, β and γ are parameters to be estimated, which can be related to the
pipeline and fluid characteristics by using other equations for head losses.
For the DW equation: γ = 1, α =
f (Q)
2φArQγ−1
, β =
gAr
Leq
.
For the HW equation: Ω =
10.67Leq
C1.852φ4.8704
, γ = 0.852.
A general formula for friction losses
It involves other head loss equation.

Nonlinear estimation of a power law for the friction in a pipeline | Identification approach
Contents
1 Motivation
2 Contribution
3 Background

Approach description
The least squares approach:
Steps
To obtain recordings (Time
Series Data) of Q and ∆Htot at
different operation points (from
the lowest to the highest
points).
To obtain the mean of each
recording: ¯Q, ¯∆Htot.
To fit ¯Q vs ∆Htot by using least
squares for instance.
0 200 400 600 800 1000 1200 1400
0
5
10
15
20
[m]
Htot
0 200 400 600 800 1000 1200 1400
[s]
2
4
6
8
10
12
[m3
/s]
10
-3
Q
2 3 4 5 6 7 8 9 10 11 12
Q [L]
0
2
4
6
8
10
12
14
16
18
20
Htot
[m]
Experimental Data
MATLAB fitted curve

Approach description
The proposed approach:
Steps
To induce steady-oscillatory flow in the pipeline provoking a
sinusoidal pressure at the upstream end.
To estimate β and γ by using a nonlinear observer.
To estimate α by using an algebraic equation or another nonlinear
observer.
∆Htot = ΩQ1+γ, where Ω =
α
β
.
Advantages
1 The time required for the estimation is shorter.
2 It is not necessary to set the pipeline at lowest and highest operation points.

OBSERVER 1
Mean
OBSERVER 2
Mean
totH
Q
Q
Q
totH
Q

 
 

tot in outH H H  
inH
outH
Q
d
dt

Nonlinear estimation of a power law for the friction in a pipeline | Design of the observers
Contents
1 Motivation
2 Contribution
3 Background

Design of the observers
Momentum equation:
˙Q =
gAr
Leq
∆Htot − Js(Q)
Js(Q) is the dissipation term that depends on the head losses formula used
(e.g. Js = f (Q)/2φArif DW equation is used). By substituting our proposed
power law in the momentum equation
˙Q = β∆Htot − αQ|Q|γ
If only positive flow is considered, then
˙Q = β∆Htot − αQ1+γ

1st Observer: β and γ estimation
By x1 = Q, x2 = β, x3 = αQγ, x4 = γ, we get
˙x(t)=




0 u1 −y 0
0 0 0 0
0 0 0 0
0 0 0 0



 x +




0
0
x3x4u2/y
0




y = 1 0 0 0 x (1)
where u1 = ∆Htot and u2 = ˙Q is the derivative of the flow rate measurement.
˙x(t) = A(u, y)x + B(u, x)
y = Cx

It has already been proven by Torres et al. 20122
that under excitation condition,
with u bounded and making A(u, y) bounded, if B(u, x) is globally Lipschitz in z
uniformly in u, one can obtain an estimation of the state for system with the
high-gain Kalman-like observer given by
˙ˆx = A(u, y)ˆx + B(u, ˆx) − Λ(λ)SCT
(u)(ˆy − y)
ˆy = Cˆx
˙S = λ(θS + [A(u, y) + dBλ(u, ˆx)]S + S[A(u, y) + dBλ(u, ˆx)]T
− SCT
CS),
with Λ(λ) =





λIN1
0
λ2
IN2
...
0 λq
INq





, dBλ =
1
λ
Λ−1
(λ)
∂B
∂x
Λ(λ), ˆx(0) ∈ RN
,
S(0) ≥ 0, which ensure
x(t) − ˆx(t) ≤ µe−σt
, µ > 0, ∀t ≥
1
λ
.
2
Torres, L., Besançon, G. and Georges, D. (2012). EKF-like observer with stability for a class
of nonlinear systems. IEEE Transactions on Automatic Control, 57(6), 1570-1574.

2nd Observer: α estimation
By defining x1 = Q, x2 = α, we get system
˙x =
0 yˆγ|y|
0 0
x +
ˆβu1
0
,
y = 1 0 x(t) = Q, (2)
where ˆβ and ˆγ represent the parameters estimated in the previous step.
u1 = ∆Htot. To estimate the states of such a system, we used an exponential
proposed by Besançon19963:
3
Besançon, G., Bornard, G., and Hammouri, H. (1996). Observer synthesis for a class of
nonlinear control systems. European Journal of control, 2(3), 176-192.

Nonlinear estimation of a power law for the friction in a pipeline | Experimental results
Contents
1 Motivation
2 Contribution
3 Background

Laboratory Pipeline Installations @ II-UNAM
Physical parameters
φ = 0.076 [m], L = 163.62 [m].

0 50 100 150 200 250 300 350 400 450
5
10
15
20
[m]
Δ Htot
0 50 100 150 200 250 300 350 400 450
[s]
6
8
10
12
[m3
/s]
×10-3
Q

0 50 100 150 200 250 300 350 400 450
[s]
-2
-1
0
1
2
3
4
×10-4
β
0 50 100 150 200 250 300 350 400 450
[s]
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
γ
0 50 100 150 200 250 300 350 400 450
[s]
0
0.5
1
1.5
2
2.5
3
α
Parameter average
¯β = 2.2941 × 10−4,
¯γ = 0.4403, ¯α = 2.5599.
Identified Momentum
Equation
˙Q = 2.56∆Htot − 0.4403Q1.44

Experimental Results
2 4 6 8 10 12
Q [L]
0
2
4
6
8
10
12
14
16
18
20
ΔH
tot
Experimental Data
Power Law
MATLAB fitted curve
3 4 5 6 7 8 9 10 11 12
Q [L] ×10-3
2
4
6
8
10
12
14
16
18
ΔH
tot
Steady State Points
Steady Oscillatory Flow

qatlho’
Danke谢谢
Grazie
Спасибо
ขอบคุณ
9C4#5$Ì
‫ﺷﻜﺮا‬ Merçi
Gracias
நன்றி
Obrigado
Ευχαριστώ
감사합니다
ध यवाद
Terima kasih
Thank you
ありがとう
Tapadh leibh
ཐུགས་རྗེ་ཆེ་།
Go raibh maith agaibh
Xin cảm ơn
Questions? ftorreso@iingen.unam.mx
http://lizeth-torres.info

Nonlinear estimation of a power law for the friction in a pipeline

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