. Integer Nonlinear Programming Approach . . . . . . . . . . Solving Heated Oil Pipeline Problems Via Mixed Yu-Hong Dai . Academy of Mathematics and Systems Science, Chinese Academy of Sciences dyh@lsec.cc.ac.cn Collaborators: Muming Yang (AMSS, CAS) Yakui Huang (Hebei University of Technology) Bo Li (PetroChina Pipeline R & D Center) CO@Work 2020 Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 40
. Introduction . . . . . . . . . Overview 1 2 . Heated Oil Pipelines Problem Formulation 3 Nonconvex and Convex Relaxations and Their Equivalence 4 The Branch-and-Bound Algorithm and Preprocessing Procedure 5 Numerical Results 6 Conclusions Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 40
. Introduction . . . . . . . . . Overview 1 2 . Heated Oil Pipelines Problem Formulation 3 Nonconvex and Convex Relaxations and Their Equivalence 4 The Branch-and-Bound Algorithm and Preprocessing Procedure 5 Numerical Results 6 Conclusions Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 / 40
. . . . . . . . . . . . . . . . Crude Oil and Pipeline Transport Figure: Pipes and stations Crude oil Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . . . . . . . . . . . . . . . . . . . . . . . . 4 / 40 ▶ is one of the most important resources ▶ can produce many kinds of fuel and chemical products ▶ is usually ( 51% around the world) transported by pipelines
. . . . . . . . . . . . . . . . Energy Loss in Pipeline Pressure P and head H of the oil Figure: Mileage-head curve (left) and mileage-temperature curve (right) Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . . . . . . . . . . . . . . . . . . . . . . . . 5 / 40 P = ρ gH where ρ is the density of the oil, g is gravity acceleration ▶ Head (pressure) loss : friction and elevation difgerence ▶ Temperature loss : dissipation
. . . . . . . . . . . . . . . Crude Oil Property Without proper transport temperature , some crude oil may Normal temperature pipelines are incapable! Figure: Viscosity-temperature curves Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 40 ▶ dramatically increase the viscosity ( high friction ) ▶ precipitate wax ▶ freeze (Some oil freezes under 32 ◦ C )
. . . . . . . . . . . . . . . . The Heated Oil Pipeline (HOP) Figure: Stations with pumps and furnaces In each station Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . . . . . . . . . . . . . 7 / 40 . . . . . . . . . . . Station Heating furnace Regulator Pumps Pipeline ▶ Heating furnace : variable ∆ T ∈ R + ▶ Constant speed pump (CSP) : constant H CP [ SP ] ▶ Shifted speed pump (SSP) : variable ∆ H SP ∈ H SP , H ▶ Regulator : head restriction
. . . . . . . . . . . . Operation Scheme and Cost . Safety requirements Figure: Temperature ( T ) and transport cost ( S ) huge cost difgerences (vary over 50,000 yuan/d) (consumes fuel equivalent to 1% transported oil) The optimal scheme will save a lot! Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . . . . . . . . . . . . . . . 8 / 40 . . . . . . . . . . . . ▶ Inlet, outlet head and temperature bounds in each station ▶ Head bounds in the pipeline ▶ Lots of feasible schemes with ▶ High heating consumption ▶ Huge rate of fmow per day (about 72,000 m 3 /d)
. . . . . . . . . . . . . . . . MINLP Implementation and the Nonconvexity Darcy-Weisbach formula and Reynold numbers ) L is the pipe length, T is the oil temperature, Q is the volume fmow of Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . . . . . . . . . . . . . . . . . . . . . 9 / 40 . . . ▶ (Nonconvex) MINLP Model ▶ Integer (binary) variables: on-ofg status of pumps ▶ Continuous variables: temperature rise comes from heating furnaces ▶ Nonconvexity: Friction loss (hydraulic friction based on HF ( T , Q , D , L ) = β ( T ) Q 2 − m ( T ) D 5 − m ( T ) ν ( T ) m ( T ) L oil, D is the inner diameter of the pipe, β ( · ) and m ( · ) are piecewise constant functions , ν ( · ) is the kinematic viscosity of oil. ▶ HF is nonconvex or even discontinuous about T in general
. . . . . . . . . . . . Motivation and Contribution . M Literatures focus on approximation or meta-heuristics C Consider using deterministic global optimization methods M Lack detailed and general mathematical model C Consider difgerent kinds of pumps and a general formulation of hydraulic frictions M General MINLP solvers may not effjcient on the HOP problems C Design an effjcient specifjc algorithm for HOP problems Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 / 40 ▶ Two-cycle strategy based on model decomposition [Wu and Yan, 1989] ▶ Improved genetic algorithm [Liu et al., 2015] ▶ Difgeretial evolution and particle swarm optimization [Zhou et al., 2015] ▶ Linear approximation [Li et al., 2011] ▶ Simulated annealing algorithm [Song and Yang, 2007]
. Introduction . . . . . . . . . Overview 1 2 . Heated Oil Pipelines Problem Formulation 3 Nonconvex and Convex Relaxations and Their Equivalence 4 The Branch-and-Bound Algorithm and Preprocessing Procedure 5 Numerical Results 6 Conclusions Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 / 40
. . . . . . . . . . . . . . . Notations Suppose . j j Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . 12 / 40 . . . . . . . . . . . . . . . . . . . . . . . ▶ there are N S stations in the pipeline; j segments, j = 1 , ..., N S − 1 ; ▶ the pipe between stations j and j + 1 is divided into N P ▶ there are N CP CSPs and N SP SSPs in station j . Heat transfer coefficient 𝑳 𝒌𝒔 𝑸 𝒌𝒔 Head 𝑰 𝒑𝒗𝒖 𝑸 𝒌𝒔:𝟐 𝑸 𝒌𝒔 Head 𝑰 𝒑𝒗𝒖 Temperature 𝑼 𝒑𝒗𝒖 Volume flow 𝑹 𝒌𝒔 𝑸 𝒌𝒔 𝑸 𝒌𝒔:𝟐 Temperature 𝑼 𝒑𝒗𝒖 Average temperature 𝑼 𝒃𝒘𝒇 Station 𝒌 + 𝟐 Friction 𝑮 𝒌𝒔 Outer diameter Pipe length 𝑴 𝒌𝒔 𝒆 𝒌𝒔 Inner diameter 𝑬 𝒌𝒔 Segment 𝒔 Elevation difference 𝚬𝐚 𝐤𝐬 Segment 𝒔 − 𝟐 Mileage difference ≈ 𝑴 𝒌𝒔 Station 𝒌 𝑫𝑸 Powered-on CSP 𝒚 𝒌 Powered-on SSP 𝒛 𝒌 SSP head 𝜠𝑰 𝒌 𝑻𝑸 Temperature rise 𝜠𝑼 𝒌 CSP head 𝑰 𝒌 𝑻 𝒌 Outlet head 𝑰 𝒑𝒗𝒖 𝑻 𝒌 𝑻 𝒌 Outlet temperature 𝑼 𝒑𝒗𝒖 𝑻 𝒌 Inlet head 𝑰 𝒋𝒐 Inlet temperature 𝑼 𝒋𝒐 Figure: Constants (blue) and variables (red) in the pipe between stations j and j + 1
. (2) . . . . . . MINLP Model for HOP Pipe calculation constraints H P jr out . (1) T P jr empirical formula ) . T P jr f out T P jr f (3) T P jr out (4) heat of the oil. Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . 13 / 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ▶ Head loss − F jr − ∆ Z jr , j = 1 , ..., N S − 1 , r = 1 , ..., N P out = H P j , r − 1 j . ▶ Friction L jr , j = 1 , ..., N S − 1 , r = 1 , ..., N P ( ) F jr = f ave , Q jr , D jr j . ▶ Average temperature (based on axial temperature drop formula and [ ( )] T P j , r − 1 − out = T P jr g + T P jr + g + T P jr e − α jr L jr , ave = 1 + 2 out , j = 1 , ..., N S − 1 , r = 1 , ..., N P 3 T P j , r − 1 j . 3 T P jr T g is the ground temperature, T f is the environment temperature variation caused by friction heat, α = ( K π d )/( ρ Qc ) is a parameter, c is the specifjc
. j . . . . MINLP Model for HOP (cont’d) Heating station calculation constraints H S j . j j (5) T S j (6) y j H SP j . SP (7) Connection constraints P jNP j in (8) P jNP j in (9) Yu-Hong Dai (AMSS, CAS) Solving HOP Problems Via MINLP CO@Work 2020 . x j H CP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 / 40 . ▶ Outlet value out , j = 1 , ..., N S − 1 , ( ) in + + ∆ H SP ≥ H S j out , j = 1 , ..., N S − 1 . in + ∆ T j = T S j Constraints (5) are inequalities due to the regulators ▶ SSP head bound j , j = 1 , . . . , N S − 1 . ≤ ∆ H SP ≤ y j H ▶ Pipe station connection , j = 1 , ..., N S − 1 , H P j 0 out = H S j +1 out = H S j out , H , j = 1 , ..., N S − 1 . T P j 0 out = T S j out = T S j +1 out , T
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