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Numerical solution to an inverse problem on a determination of places and capacities of sources in the hyperbolic systems
1. | Baku State University, Z. Khalilov, 23, Baku, Azerbaijan |
2. | Institute of Control Systems of Azerbaijan, National Academy of Sciences, B. Vahabzade, 9, AZ1141 Baku, Azerbaijan |
In the work we investigate the numerical solution to a class of inverse problems with respect to the system of differential equations of hyperbolic type. The specialties of considered problems are: 1) the impulse impacts are present in the system and it is necessary to determine the capacities and the place of their location; 2) the differential equations of the system are only related to boundary values, and arbitrarily; 3) because of the long duration of the object functioning, the exact values of the initial conditions are not known, but a set of possible values is given. The inverse problem under consideration is reduced to the problem of parametric optimal control without initial conditions with non-separated boundary conditions. For the solution it is proposed to use first-order optimization methods. The results of numerical experiments are given on the example of the inverse problem of fluid transportation in the pipeline networks of complex structure. The problem is to determine the locations and the volume of leakage of raw materials based on the results of additional observations of the state of the transportation process at internal points or at the ends of sections of the pipeline network.
References:
[1] |
A. Adamkowski,
Analysis of transient flow in pipes with expanding or contracting sections, J. Fluids Eng., 125 (2003), 716-722.
|
[2] |
K. R. Aida-zade,
Computational problems on hydraulic networks, Computational Math. and Math. Physics, 29 (1989), 125-132.
doi: 10.1016/0041-5553(89)90056-6. |
[3] |
K. R. Aida-zade and J. A. Asadova,
Study of transients in oil pipelines, Aut. and Remote Control, 72 (2011), 2563-2577.
|
[4] |
K. R. Aida-zade and E. R. Ashrafova,
Localization of the points of leakage in an oil main pipeline under non-stationary conditions, J. of Eng. Phys. and Therm., 85 (2012), 1148-1156.
|
[5] |
K. R. Aida-zade and Y. R. Ashrafova,
Calculation of transient fluid flow regimes in pipeline networks, Sib. Zh. Ind. Mat., 18 (2015), 12-23.
doi: 10.17377/sibjim.2015.18.202. |
[6] |
K. R. Aida-zade and Y. R. Ashrafova,
Optimal control of sources on some classes of functions, Optimization: A J. of Mathem. Prog. and Oper. Research., 63 (2014), 1135-1152.
doi: 10.1080/02331934.2012.711831. |
[7] |
K. R. Aida-zade and Y. R. Ashrafova,
Solving systems of differential equations of block structure with nonseparated boundary conditions, J. of Appl. and Industrial Math., 9 (2015), 1-10.
doi: 10.1134/S1990478915010019. |
[8] |
K. R. Aida-zade and A. V. Handzel,
An approach to lumped control synthesis in distributed systems, Appl. and Computational Math., 6 (2007), 69-79.
|
[9] |
A. M. Al-Khomairi,
Leak detection in long pipelines using the least squares method, J. Hydraulic Res., 46 (2008), 392-401.
|
[10] |
Y. R. Ashrafova,
Numerical investigation of the duration of the effect exerted by initial regimes on the process of liquid motion in a pipeline, J. of Eng. Physics and Therm., 88 (2015), 1-9.
|
[11] |
K. R. Ayda-zade and V. M. Abdullayev,
Numerical solution of optimal control problems with unseparated conditions on phase state, Appl. and Computational Math.: An Int. Journal, 4 (2005), 165-177.
|
[12] |
A. Bergant, A. R. Simpson and J. P. Vitkovsky,
Developments in unsteady pipe flow friction modeling, J. Hydraulic Res. IAHR, 39 (2001), 249-257.
|
[13] |
A. Q. Butkovskii and L. M. Pustilnikov, Theory of moving control of systems with distributed parameters, in Series in Theoretical Foundations of Engineering Cybernetics, Moscow, 1980. |
[14] |
X. Chao, D. Yimeng, R. Zhigang, J. Huachen and Y. Xin,
Sensor deployment for pipeline leakage detection via optimal boundary control strategies, J. of Indust. and Mgmt. Optimization, 11 (2015), 199-216.
doi: 10.3934/jimo.2015.11.199. |
[15] |
I. A. Charniy, Unsteady motion of a real fluid in pipes, Moskva, Nedra, 1975. |
[16] |
M. H. Chaudhry, Applied hydraulic transients, Van Nostrand Reinhold, New York, 1987. |
[17] |
T. Chen, Z. Ren, C. Xu and R. Loxton,
Optimal boundary control for water hammer suppression in fluid transmission pipelines, Comp. & Math. with Appl., 69 (2015), 275-290.
doi: 10.1016/j.camwa.2014.11.008. |
[18] |
T. Chen, C. Xu, Q. Lin, et al., Water hammer mitigation via PDE-constrained optimization, J. Control Engineering Practice, 45 (2015), 54-63. |
[19] |
A. F. Colombo, P. Lee and B. W. Karney,
A selective literature review of transient-based leak detection methods, J. Hydro-environment Res., 2 (2009), 212-227.
|
[20] |
S. Datta and S. Sarkar,
A review on different pipeline fault detection methods, J. of Loss Prevention in the Process Indust., 41 (2016), 97-106.
|
[21] |
M. Ferrante, B. Brunone, S. Meniconi, B. W. Karney and and C. Massari,
Leak Size, Detectability and Test Conditions in Pressurized Pipe Systems, Water Resources Mgmt., 28 (2014), 4583-4598.
|
[22] |
Z. Guanghui, N. Qin and Z. Meilan,
A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems, J. of Indust. and Mgmt. Optimization, 13 (2017), 595-608.
doi: 10.3934/jimo.2016034. |
[23] |
M. A. Huseynzade and V. A. Yufin, Unsteady oil and gas movement in main pipelines, M.: Nedra, 1981,232p. |
[24] |
Z. S. Kapelan, D. A. Savic and G. A. Walters,
A hybrid inverse transient model for leakage detection and roughness calibration in pipe networks, J. Hydraulic Res., 41 (2003), 481-492.
|
[25] |
O. A. Ladijenskaya, Boundary-value probems of mathematical physics, Moskva, Nauka, 1973. |
[26] |
H. G. Lee and J. Kim,
Regularized Dirac delta functions for phase field models, Int. J. for Num. Methods in Engineering, 91 (2012), 269-288.
|
[27] |
K. F. Oyedeko and H. A. Balogun, Modeling and simulation of a leak detection for oil and gas pipelines via transient model: A case study of the niger delta, J. of Energy Tech. and Policy, 5 (2015). |
[28] |
A. A. Samarskii, Theory of difference schemes, Moskow, Nauka, 1989. |
[29] |
A. A. Samarskii and P. N. Vabishevich, Numerical method to solve inverse problems of mathematical physics, Moskva, LKI, 2009. |
[30] |
H. Shamloo and A. Haghighi,
Leak detection in pipelines by inverse backward transient analysis, J. Hydraulic Res., 47 (2009), 311-318.
|
[31] |
H. Shamloo and A. Haghighi,
Optimum leak detection and calibration of pipe networks by inverse transient analysis, J. Hydraulic Res., 48 (2010), 371-376.
doi: 10.1080/00221681003726304. |
[32] |
R. Wichowski,
Hydraulic transients analysis in pipe networks by the method of characteristics (MOC), Archives of Hydro-Engineering and Environ. Mech., 53 (2006), 267-291.
|
[33] |
E. B. Wylie and V. L. Streeter, Fluid Transients, McGraw-Hill International Book Co., New York, 1978. |
show all references
References:
[1] |
A. Adamkowski,
Analysis of transient flow in pipes with expanding or contracting sections, J. Fluids Eng., 125 (2003), 716-722.
|
[2] |
K. R. Aida-zade,
Computational problems on hydraulic networks, Computational Math. and Math. Physics, 29 (1989), 125-132.
doi: 10.1016/0041-5553(89)90056-6. |
[3] |
K. R. Aida-zade and J. A. Asadova,
Study of transients in oil pipelines, Aut. and Remote Control, 72 (2011), 2563-2577.
|
[4] |
K. R. Aida-zade and E. R. Ashrafova,
Localization of the points of leakage in an oil main pipeline under non-stationary conditions, J. of Eng. Phys. and Therm., 85 (2012), 1148-1156.
|
[5] |
K. R. Aida-zade and Y. R. Ashrafova,
Calculation of transient fluid flow regimes in pipeline networks, Sib. Zh. Ind. Mat., 18 (2015), 12-23.
doi: 10.17377/sibjim.2015.18.202. |
[6] |
K. R. Aida-zade and Y. R. Ashrafova,
Optimal control of sources on some classes of functions, Optimization: A J. of Mathem. Prog. and Oper. Research., 63 (2014), 1135-1152.
doi: 10.1080/02331934.2012.711831. |
[7] |
K. R. Aida-zade and Y. R. Ashrafova,
Solving systems of differential equations of block structure with nonseparated boundary conditions, J. of Appl. and Industrial Math., 9 (2015), 1-10.
doi: 10.1134/S1990478915010019. |
[8] |
K. R. Aida-zade and A. V. Handzel,
An approach to lumped control synthesis in distributed systems, Appl. and Computational Math., 6 (2007), 69-79.
|
[9] |
A. M. Al-Khomairi,
Leak detection in long pipelines using the least squares method, J. Hydraulic Res., 46 (2008), 392-401.
|
[10] |
Y. R. Ashrafova,
Numerical investigation of the duration of the effect exerted by initial regimes on the process of liquid motion in a pipeline, J. of Eng. Physics and Therm., 88 (2015), 1-9.
|
[11] |
K. R. Ayda-zade and V. M. Abdullayev,
Numerical solution of optimal control problems with unseparated conditions on phase state, Appl. and Computational Math.: An Int. Journal, 4 (2005), 165-177.
|
[12] |
A. Bergant, A. R. Simpson and J. P. Vitkovsky,
Developments in unsteady pipe flow friction modeling, J. Hydraulic Res. IAHR, 39 (2001), 249-257.
|
[13] |
A. Q. Butkovskii and L. M. Pustilnikov, Theory of moving control of systems with distributed parameters, in Series in Theoretical Foundations of Engineering Cybernetics, Moscow, 1980. |
[14] |
X. Chao, D. Yimeng, R. Zhigang, J. Huachen and Y. Xin,
Sensor deployment for pipeline leakage detection via optimal boundary control strategies, J. of Indust. and Mgmt. Optimization, 11 (2015), 199-216.
doi: 10.3934/jimo.2015.11.199. |
[15] |
I. A. Charniy, Unsteady motion of a real fluid in pipes, Moskva, Nedra, 1975. |
[16] |
M. H. Chaudhry, Applied hydraulic transients, Van Nostrand Reinhold, New York, 1987. |
[17] |
T. Chen, Z. Ren, C. Xu and R. Loxton,
Optimal boundary control for water hammer suppression in fluid transmission pipelines, Comp. & Math. with Appl., 69 (2015), 275-290.
doi: 10.1016/j.camwa.2014.11.008. |
[18] |
T. Chen, C. Xu, Q. Lin, et al., Water hammer mitigation via PDE-constrained optimization, J. Control Engineering Practice, 45 (2015), 54-63. |
[19] |
A. F. Colombo, P. Lee and B. W. Karney,
A selective literature review of transient-based leak detection methods, J. Hydro-environment Res., 2 (2009), 212-227.
|
[20] |
S. Datta and S. Sarkar,
A review on different pipeline fault detection methods, J. of Loss Prevention in the Process Indust., 41 (2016), 97-106.
|
[21] |
M. Ferrante, B. Brunone, S. Meniconi, B. W. Karney and and C. Massari,
Leak Size, Detectability and Test Conditions in Pressurized Pipe Systems, Water Resources Mgmt., 28 (2014), 4583-4598.
|
[22] |
Z. Guanghui, N. Qin and Z. Meilan,
A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems, J. of Indust. and Mgmt. Optimization, 13 (2017), 595-608.
doi: 10.3934/jimo.2016034. |
[23] |
M. A. Huseynzade and V. A. Yufin, Unsteady oil and gas movement in main pipelines, M.: Nedra, 1981,232p. |
[24] |
Z. S. Kapelan, D. A. Savic and G. A. Walters,
A hybrid inverse transient model for leakage detection and roughness calibration in pipe networks, J. Hydraulic Res., 41 (2003), 481-492.
|
[25] |
O. A. Ladijenskaya, Boundary-value probems of mathematical physics, Moskva, Nauka, 1973. |
[26] |
H. G. Lee and J. Kim,
Regularized Dirac delta functions for phase field models, Int. J. for Num. Methods in Engineering, 91 (2012), 269-288.
|
[27] |
K. F. Oyedeko and H. A. Balogun, Modeling and simulation of a leak detection for oil and gas pipelines via transient model: A case study of the niger delta, J. of Energy Tech. and Policy, 5 (2015). |
[28] |
A. A. Samarskii, Theory of difference schemes, Moskow, Nauka, 1989. |
[29] |
A. A. Samarskii and P. N. Vabishevich, Numerical method to solve inverse problems of mathematical physics, Moskva, LKI, 2009. |
[30] |
H. Shamloo and A. Haghighi,
Leak detection in pipelines by inverse backward transient analysis, J. Hydraulic Res., 47 (2009), 311-318.
|
[31] |
H. Shamloo and A. Haghighi,
Optimum leak detection and calibration of pipe networks by inverse transient analysis, J. Hydraulic Res., 48 (2010), 371-376.
doi: 10.1080/00221681003726304. |
[32] |
R. Wichowski,
Hydraulic transients analysis in pipe networks by the method of characteristics (MOC), Archives of Hydro-Engineering and Environ. Mech., 53 (2006), 267-291.
|
[33] |
E. B. Wylie and V. L. Streeter, Fluid Transients, McGraw-Hill International Book Co., New York, 1978. |



60 | 20 | 90 | 10 | 45,685 | ||
76.104 | 16.704 | 11.664 | 42.48 | 57.636 | ||
5.73 |
1.26 |
3.19 |
1.85 |
7.43 |
||
0% | 30.003 | 29.998 | 30.008 | 29.994 | 29.998 | |
6 | 5 | 16 | 14 | 8 | ||
0.00009 | 0.00006 | 0.0003 | 0.0002 | 0.00006 | ||
0.0003 | 0. 0006 | 0.0003 | 0.0002 | 0.0003 | ||
76.352 | 16.837 | 11.683 | 42.148 | 57.732 | ||
0.023 | 0.014 | 0.024 | 0.017 | 0.020 | ||
0.5% | 29.841 | 30.332 | 30.068 | 29.654 | 29.796 | |
6 | 5 | 14 | 12 | 7 | ||
–0.005 | 0.011 | 0.002 | –0.011 | –0.006 | ||
0.043 | 0.030 | 0.049 | 0.041 | 0.042 | ||
77.119 | 16.924 | 11.832 | 43.744 | 57.413 | ||
0.067 | 0.071 | 0.073 | 0.062 | 0.065 | ||
1% | 28.527 | 29.392 | 29.923 | 30.597 | 29.839 | |
6 | 5 | 14 | 12 | 7 | ||
–0.049 | –0.020 | –0.002 | 0.020 | –0.005 | ||
0.109 | 0.092 | 0.107 | 0.094 | 0.093 |
60 | 20 | 90 | 10 | 45,685 | ||
76.104 | 16.704 | 11.664 | 42.48 | 57.636 | ||
5.73 |
1.26 |
3.19 |
1.85 |
7.43 |
||
0% | 30.003 | 29.998 | 30.008 | 29.994 | 29.998 | |
6 | 5 | 16 | 14 | 8 | ||
0.00009 | 0.00006 | 0.0003 | 0.0002 | 0.00006 | ||
0.0003 | 0. 0006 | 0.0003 | 0.0002 | 0.0003 | ||
76.352 | 16.837 | 11.683 | 42.148 | 57.732 | ||
0.023 | 0.014 | 0.024 | 0.017 | 0.020 | ||
0.5% | 29.841 | 30.332 | 30.068 | 29.654 | 29.796 | |
6 | 5 | 14 | 12 | 7 | ||
–0.005 | 0.011 | 0.002 | –0.011 | –0.006 | ||
0.043 | 0.030 | 0.049 | 0.041 | 0.042 | ||
77.119 | 16.924 | 11.832 | 43.744 | 57.413 | ||
0.067 | 0.071 | 0.073 | 0.062 | 0.065 | ||
1% | 28.527 | 29.392 | 29.923 | 30.597 | 29.839 | |
6 | 5 | 14 | 12 | 7 | ||
–0.049 | –0.020 | –0.002 | 0.020 | –0.005 | ||
0.109 | 0.092 | 0.107 | 0.094 | 0.093 |
60 | 20 | 90 | 10 | 45.685 | ||
155.248 | 22.323 | 22.323 | 84.303 | 118.809 | ||
8.2 |
2.7 |
1.0 |
2.2 |
7.4 |
||
3 | 30.145 | 29.996 | 30.190 | 30.022 | 30.029 | |
0.004 | 0.0001 | 0.006 | 0.0007 | 0.0005 | ||
0.0005 | 0.0004 | 0.0001 | 0.0009 | 0.0005 | ||
155.390 | 33.608 | 22.443 | 84.086 | 118.854 | ||
1.6 |
3.9 |
4.3 |
2.8 |
6.4 |
||
5 | 29.963 | 30.093 | 30.100 | 29.985 | 29.995 | |
0.001 | 0.003 | 0.003 | 0.0005 | 0.0001 | ||
0.0006 | 0.0003 | 0.0005 | 0.0004 | 0.0002 |
60 | 20 | 90 | 10 | 45.685 | ||
155.248 | 22.323 | 22.323 | 84.303 | 118.809 | ||
8.2 |
2.7 |
1.0 |
2.2 |
7.4 |
||
3 | 30.145 | 29.996 | 30.190 | 30.022 | 30.029 | |
0.004 | 0.0001 | 0.006 | 0.0007 | 0.0005 | ||
0.0005 | 0.0004 | 0.0001 | 0.0009 | 0.0005 | ||
155.390 | 33.608 | 22.443 | 84.086 | 118.854 | ||
1.6 |
3.9 |
4.3 |
2.8 |
6.4 |
||
5 | 29.963 | 30.093 | 30.100 | 29.985 | 29.995 | |
0.001 | 0.003 | 0.003 | 0.0005 | 0.0001 | ||
0.0006 | 0.0003 | 0.0005 | 0.0004 | 0.0002 |
25 | 15.219 | 2.556 | 3.55 |
165 | |
10 | 14.781 | 44.615 | 3.55 |
204 | |
22.56 | 15.222 | 3.467 | 3.59 |
197 |
25 | 15.219 | 2.556 | 3.55 |
165 | |
10 | 14.781 | 44.615 | 3.55 |
204 | |
22.56 | 15.222 | 3.467 | 3.59 |
197 |
Segment | Number of | |||||
indices |
iterations | |||||
(1, 2) | 5.991 | 43.272 | 39.614 | 6 | ||
(5, 2) | 2.279 | 139.536 | 60.386 | 58 | ||
25 | (3, 2) | 29.974 | 10.188 | 1.22 |
12 | |
(5, 4) | 2.244 | 54.185 | 37.958 | 79 | ||
(5, 6) | 2.995 | 72.396 | 20.621 | 185 | ||
(1, 2) | 5.986 | 46.836 | 39.614 | 5 | ||
(5, 2) | 2.305 | 84.132 | 60.408 | 5 | ||
5 | (3, 2) | 29.935 | 32.148 | 2.44 |
12 | |
(5, 4) | 1.898 | 38.628 | 37.728 | 150 | ||
(5, 6) | 2.947 | 21.786 | 20.621 | 72 |
Segment | Number of | |||||
indices |
iterations | |||||
(1, 2) | 5.991 | 43.272 | 39.614 | 6 | ||
(5, 2) | 2.279 | 139.536 | 60.386 | 58 | ||
25 | (3, 2) | 29.974 | 10.188 | 1.22 |
12 | |
(5, 4) | 2.244 | 54.185 | 37.958 | 79 | ||
(5, 6) | 2.995 | 72.396 | 20.621 | 185 | ||
(1, 2) | 5.986 | 46.836 | 39.614 | 5 | ||
(5, 2) | 2.305 | 84.132 | 60.408 | 5 | ||
5 | (3, 2) | 29.935 | 32.148 | 2.44 |
12 | |
(5, 4) | 1.898 | 38.628 | 37.728 | 150 | ||
(5, 6) | 2.947 | 21.786 | 20.621 | 72 |
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