Optimal Control of Transient Flows in Pipeline Networks with Heterogeneous Mixtures of Hydrogen and Natural Gas Luke S. Baker1 Saif R. Kazi2 Rodrigo B. Platte1 and Anatoly Zlotnik2

2025-04-26 0 0 925.75KB 8 页 10玖币

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Optimal Control of Transient Flows in Pipeline Networks with

Heterogeneous Mixtures of Hydrogen and Natural Gas

Luke S. Baker1, Saif R. Kazi2, Rodrigo B. Platte1, and Anatoly Zlotnik2

Abstract— We formulate a control system model for the

distributed ﬂow of mixtures of highly heterogeneous gases

through large-scale pipeline networks with time-varying in-

jections of constituents, withdrawals, and control actions of

compressors. This study is motivated by the proposed blending

of clean hydrogen into natural gas pipelines as an interim

means to reducing end use carbon emissions while utilizing

existing infrastructure for its planned lifetime. We reformulate

the partial differential equations for gas dynamics on pipelines

and balance conditions at junctions using lumped elements to a

sparse nonlinear differential algebraic equation system. Our key

advance is modeling the mixing of constituents in time through-

out the network, which requires doubling the state space needed

for a single gas and increases numerical ill-conditioning. The

reduced model is shown to be a consistent approximation of

the original system, which we use as the dynamic constraints

in a model-predictive optimal control problem for minimizing

the energy expended by applying time-varying compressor

operating proﬁles to guarantee time-varying delivery proﬁles

subject to system pressure limits. The optimal control problem

is implemented after time discretization using a nonlinear

program, with validation of the results done using a transient

simulation. We demonstrate the methodology for a small test

network, and discuss scalability and potential applications.

I. INTRODUCTION

Transportation of natural gas through networks of large-

scale transmission pipelines has been studied in steady-state

[1], [2], [3], [4], [5] and transient [6], [7], [8] operations

with applications to the optimal control of compressor ac-

tuators. In steady-state, the ﬂow of gas in the network is

balanced, so that inﬂows from processing plants and supply

stations and outﬂows from withdrawal stations sum to zero.

Steady-state pipeline ﬂows are described using simple time-

invariant algebraic equations that relate pressure drop in

the direction of ﬂow to mass ﬂow along each pipeline.

In the transient regime, computational complexity increases

signiﬁcantly because the ﬂow in each pipeline cannot be

modeled with simple algebraic equations but rather requires

a system of nonlinear partial differential equations (PDEs)

*This study was supported by the U.S. Department of Energy’s Advanced

Grid Modeling (AGM) project “Dynamical Modeling, Estimation, and

Optimal Control of Electrical Grid-Natural Gas Transmission Systems”,

as well as LANL Laboratory Directed R&D project “Efﬁcient Multi-scale

Modeling of Clean Hydrogen Blending in Large Natural Gas Pipelines to

Reduce Carbon Emissions”. Research conducted at Los Alamos National

Laboratory is done under the auspices of the National Nuclear Security

Administration of the U.S. Department of Energy under Contract No.

89233218CNA000001.

1Luke Baker and Rodrigo Platte are with the School of Mathematical

and Statistical Sciences at Arizona State University, Tempe, Arizona, 85281;

{lsbaker1,rplatte}@asu.edu.

2Saif Kazi and Anatoly Zlotnik are in the Applied Mathematics & Plasma

Physics Group at Los Alamos National Laboratory, Los Alamos, New

Mexico, 87545; {skazi,azlotnik}@lanl.gov.

[9], [10]. Model reduction methods have been proposed to

reduce the complexity of optimizing gas ﬂows in networks

[11], [12]. Although natural gas is projected to be a primary

fuel source through the year 2050 [13], worldwide economies

have invested in transitioning from fossil fuels such as natural

gas to more sustainable resources. Hydrogen is a potential

candidate, which, because it does not produce carbon dioxide

when burned, is considered to have the potential to address

climate change [14]. Natural gas pipeline operation and man-

agement protocols may be modiﬁed to transport mixtures of

natural gas and hydrogen. Recent studies indicate that natural

gas pipelines can safely and effectively transport mixtures

of up to 20% hydrogen by volume [15], [16]. However, the

complexity of modeling steady-state and transient ﬂows, and

thus designing and operating pipelines, is compounded with

the injection of hydrogen [17], [18].

Natural gas and hydrogen have signiﬁcantly different

physical and chemical properties. Hydrogen is less dense

than natural gas, and the speed of sound through hydrogen is

roughly four times as large as that of natural gas. Viscosity,

velocity, density, pressure, and energy of the gas mixture

all vary with varying hydrogen concentration [19], [20], and

these directly affect the transportation of the mixture [21].

Numerical simulations have been performed to demonstrate

various effects on steady-state and transient-state ﬂows of

mixtures of hydrogen and natural gas in pipeline networks

[22], [23], [24], [25], [26], [27], [28]. The method of

characteristics was used in the numerical simulation of

transient ﬂows on cycle networks with homogeneous ﬂow

mixtures [25]. Another recent study investigates gas compo-

sition tracking using a moving grid method and an implicit

backward difference method [23]. It was shown that both

methods of tracking perform well, but the implicit difference

method may lose some ﬁner detail in the response due to nu-

merical diffusion. A ﬁnite element method using COMSOL

Multiphysics was also developed [26]. That study considers

the effects of hydrogen concentration on the compressibility

factor of the mixture and its relation with pressure. Moreover,

there the authors demonstrate that pressure may exceed

pipeline limitations in the transient evolution of ﬂow and

that the likelihood of this happening increases proportionally

with increasing hydrogen concentration.

In contrast to pure natural gas, few studies have exam-

ined optimization of steady-state and transient operations

of mixtures of hydrogen and natural gas in networks. To

our knowledge, there are no results on the optimal control

of transient ﬂows of heterogeneous mixtures of gases in

pipelines or networks of pipelines. Optimal control of com-

arXiv:2210.06269v2 [math.OC] 22 May 2023

pressor actuators for transport of pure natural gas typically

seeks to minimize the cost of running compressors while

being subjected to PDE ﬂow dynamics, nodal pressure and

nodal ﬂow balance constraints, and inequality box constraints

that limit the pressure throughout the network [29]. Other

formulations may use an objective function that maximizes

economic value [30]. When transients are sufﬁciently slow,

a friction-dominated approximation may be made [31], and

this was shown to be valid in the regime of normal pipeline

operations [32]. We use friction-dominated modeling to sim-

plify the reduced modeling in the heterogeneous gas setting.

In this study, we formulate a control system model

for transporting heterogeneous mixtures of gases through

pipeline networks of general form, and extend optimal

control problems for gas pipeline ﬂow to this setting. Our

key advance is modeling the mixing of constituents in time

throughout the network, which requires doubling the state

space needed for a single gas and increases numerical ill-

conditioning. This enables the formulation and solution of

optimal control problems in which constituent gases may

be injected at different points in the network at varying

concentrations, e.g., the addition of 100% hydrogen at certain

nodes. An algorithm is implemented to obtain solutions, and

the results are demonstrated on a small test network that

includes a cycle.

The remainder of the manuscript is organized as follows.

The governing equations for the ﬂow of mixtures of gases

in a network are presented in Section II. In Section III,

an endpoint discretization method is employed to reduce

the system of PDEs to a system of ordinary differential

equations (ODEs). There, we show that the discretization

method is consistent and results in the equations for natural

gas only in the case of zero hydrogen injection, and yields

the steady-state equations when supply and withdrawal are

held constant. Section IV describes time-discretization of

the optimal control problem that yields a nonlinear program

(NLP). The NLP is solved for a test network in Section V,

and we discuss applications in Section VI.

II. NETWORK FLOW CONTROL FORMULATION

We begin by deﬁning notation that will be used in the

study. A gas network is modeled as a connected and di-

rected graph (E,V)that consists of edges (pipelines) E=

{1, . . . , E}and nodes (junctions) V={1,2, . . . , V }, where

Eand Vdenote the cardinalities of the sets. It is assumed

that the nodes and edges are ordered within their sets

according to their integer labels. The symbol kis reserved

for identifying edges in Eand the symbols iand jare

reserved for identifying nodes in V. Supply nodes Vs⊂ V

and withdrawal nodes Vw⊂ V are assumed to be disjoint

sets that partition V, i.e., Vs∪ Vw=Vand Vs∩ Vw=∅.

It is assumed that supply nodes are ordered in Vbefore

withdrawal nodes so that i < j for all i∈ Vsand j∈ Vw.

The graph is directed by judiciously assigning a positive ﬂow

direction along each edge. It is assumed that gas ﬂows in the

positive oriented direction of an edge so that the mass ﬂux

and velocity of the gas are positive quantities. The notation

k:i7→ jmeans that edge k∈ E is directed from node

i∈ V to node j∈ V. For each node j∈ V, we deﬁne

(potentially empty) incoming and outgoing sets of pipelines

by 7→j={k∈ E|k:i7→ j}and j7→ ={k∈ E|k:j7→ i},

respectively.

The transportation of the mixture of hydrogen and natural

gas is modeled as a simpliﬁcation of the isothermal Euler

equations. For each pipe k∈ E, the ﬂow variables are

natural gas density ρ(1)

k(t, x), hydrogen density ρ(2)

k(t, x),

and mass ﬂux ϕk(t, x)of the mixture, with t∈[0, T ]

and x∈[0, `k], where Tdenotes the time horizon and `k

denotes the length of the pipe. Assuming zero inclination

and sufﬁciently slow transients, the ﬂow through edge kis

governed by the friction-dominated PDEs

∂tρ(m)

k+∂x ρ(m)

ρ(1)

k+ρ(2)

ϕk!= 0,(1)

∂xσ2

1ρ(1)

k+σ2

2ρ(2)

k=−λk

2Dk

ϕk|ϕk|

ρ(1)

k+ρ(2)

,(2)

where (1) is deﬁned for each constituent m= 1 and m= 2.

Superscripts “1” and “2” attached to a gas variable are

conserved for identifying natural gas and hydrogen variables,

respectively. The parameters for each pipeline k∈ E are di-

ameter Dk, friction factor λk, speed of sound through natural

gas σ1, and speed of sound through hydrogen gas σ2. In the

above dynamic equations, the compressibility factors of the

gasses are assumed to be constants so that the equations of

states are ideally given by p(m)

k=σ2

mρ(m)

k, where p(m)

kis the

partial pressure. The summation of partial pressures results

in the equation of state pk= (σ2

1η(1)

k+σ2

2η(2)

k)ρk, where

pk= (p(1)

k+p(2)

k)is the total pressure, ρk= (ρ(1)

k+ρ(2)

is the total density, η(1)

k=ρ(1)

k/ρkis the concentration

of natural gas, and η(2)

k=ρ(2)

k/ρkis the concentration of

hydrogen. The concentration as deﬁned here refers to mass

fraction, so that the volumetric fraction of hydrogen in the

mixture is substantially greater.

Friction forces between the interior wall of a pipe and

gas ﬂowing through it cause pressure to decrease in the

direction of ﬂow, as reﬂected in the momentum equation

(2). Compressor stations receive gas at low pressure and

reduce its volume to increase its pressure to levels required

for transportation and customer contracts. In addition to

compressors, regulators are installed to reduce the pressure

of the received gas to within limits that are compatible

with lower pressure distribution systems. For convenience,

we assume that a compressor is located at the inlet and

a regulator is located at the outlet of each pipeline with

respect to the prescribed positive ﬂow direction. For each

pipeline k∈ E, compression and regulation are modeled

with time-varying multiplicative compressor ratio µk(t)≥1

and regulator ratio µk(t)≥1, with orientations illustrated in

Figure 1.

Natural gas and hydrogen are injected into the network

at each supply node i∈ Vswith speciﬁed time-varying

proﬁles of natural gas density s(1)

i(t)and hydrogen density

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OptimalControlofTransientFlowsinPipelineNetworkswithHeterogeneousMixturesofHydrogenandNaturalGasLukeS.Baker1,SaifR.Kazi2,RodrigoB.Platte1,andAnatolyZlotnik2AbstractWeformulateacontrolsystemmodelforthedistributedowofmixturesofhighlyheterogeneousgasesthroughlarge-scalepipelinenetworkswithtime-varyin...

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Optimal Control of Transient Flows in Pipeline Networks with Heterogeneous Mixtures of Hydrogen and Natural Gas Luke S. Baker1 Saif R. Kazi2 Rodrigo B. Platte1 and Anatoly Zlotnik2

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