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Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA
Practical Mathematical Modeling for Simulation,
Estimation, and Optimal Control of Gas Pipeline Systems
Anatoly Zlotnik
July 22, 2020
Department of Mathematics
Friedrich-Alexander-Universität Erlangen-Nürnberg
LA-UR-20-25167
Los Alamos National Laboratory
7/21/2020 | 2Los Alamos National Laboratory
• Federally Funded Research and Development Center
• Operated by National Nuclear Security Administration of U.S. Department of Energy
• Solve complex, interdisciplinary, multi-physics problems
• Advanced Network Science Initiative – https://lanl-ansi.github.io/
– Interdisciplinary team with expertise in physics, applied math, statistics, optimization, electrical and
mechanical engineering, computer science, software development, and cloud computing
– Theoretical and algorithm development for complex infrastructure optimization and control problems
– Provide third-party, independent, science-based input into complex problems of national concern
• Extensive reach back to other science-based capabilities
– Space science and space weather
– Earth and environmental science
– Chemistry and biology
– Extreme physics and effects
Collaboration
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Scott
BackhausRussell
Bent
Michael
Chertkov
Conrado
Borraz-SanchezSidhant
MisraHarsha
NagarajanMarc
Vuffray
Line
Roald
Alex
Korotkevich
Sergey
Dyachenko
Hassan
Hijazi
Pascal Van
Hentenryck
Terrence
Mak
Alex
Rudkevich
Richard
Tabors
Pablo
RuizMichael
Caramanis
Antonio
Conejo
Fei
Wu
Bining
Zhao
Ramteen
Sioshansi
Xindi
Li
Richard
Carter
Daniel
Baldwin
Anthony
Giacomoni
Russ
Philbrick
John
Goldis
Kaarthik
Sundar
Outline
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• Energy infrastructure challenges
• Inspiration from the power grid
• Modeling physics & engineering of gas pipelines for large-scale control
• Gas market design using pipeline optimization
• State and parameter estimation
• Monotonicity properties and modeling implications
• Coordination of electricity and gas transmission
Energy infrastructure challenges
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Grid modernization
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• Distributed generation, microgrids
• Increasing penetration of clean, renewable energy (20% renewables by 2030)
• New methods for automatic and responsive grid control – “smart grid”
Electricity production today
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• Electricity production by source in the United States (2019)
– Gas: 38.4%, coal: 23.5%, nuclear: 19.7%,
– Renewable 17.5% (wind: 7.3%, solar 1.8%)
• Significant construction of natural gas-fired power plants
(Source: US EIA)
Filling the demand curve
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• Gas-fired generation is used to fill the demand curve
• “Duck curve” in areas with high solar penetration
• Requires gas-fired generators to ramp up production quickly
Energy systems are now coupled
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• Power & gas transmission
infrastructures are
coupled through gas
generators
• Gas pipeline loads are
Increasing, and becoming
more variable/intermittent
• The coupling is
strengthening, as seen in
simultaneous price spikes
(ISO New England)
• Electricity production by source in
Germany (2019)
– Coal: 29%, natural gas: 10%, nuclear: 14%.
– Wind: 25%, solar: 9.1%, biomass: 8.7%,
hydroelectricity: 3.7%.
(Source: Fraunhofer, US EIA)
Generation Fuel Mix in Germany
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• Electricity production by source in European Union (2017)
– Coal: 20%, natural gas: 21%, nuclear: 25%.
– Wind: 11%, solar: 4%, biomass: 6%, hydroelectricity: 10%.
(Source: Eurostat)
Generation Fuel Mix in European Union
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Energy systems are now coupled
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• Power & gas transmission infrastructures are coupled through gas generators
• Gas pipeline loads are Increasing, and becoming more variable/intermittent
High Voltage Electricity Transmission High Pressure Natural Gas Transport
Gas pipeline operations
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• Natural Gas is traded in regulated markets
– Bilateral transactions between buyers & sellers for steady
ratable flows
• Transmission pipelines sell gas transportation to
shippers (buyers and sellers)
– Marketing and scheduling is time-consuming, not optimized
– Human operators manage fragmented systems reactively,
in real-time
– Business processes are daily, not hourly
– Business and operating standards vary by company
• Gas delivery may not adjust in real time
– Possible disparity between scheduled and actual gas flows
and pressures in normal operations
– Limited ability to react to unplanned contingencies
Inspiration from the power grid
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Power grid basics
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• Real-time pricing in electricity markets
– Optimal power flow provides generation setpoints
– Market administered for a geographic footprint
– Shadow prices are posted as real-time prices
– Locational Marginal Prices (LMPs) for electricity
• Gas pipeline management is less responsive
– Separate companies in the same geographic footprint
– Operations typically do not use optimization
• Goal: Locational Trade Values (LTVs) for natural gas
– Nodal pricing of natural gas delivery over a pipeline network
– Obtained by single price two-sided auction mechanism
– Account for pipeline structure, physics and engineering
– Generate hourly updates
Motivation
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$800
$6
PJM Interconnection price per MWh
July 19, 2013 heat wave
Motivation
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• Model-predictive optimal control of gas pipelines
– Old paradigm: Given predicted flow profiles, how to operate compressors such that
pressure remains within set limits (if possible)?
– New paradigm: Given price/quantity bids of shippers, what is the best allocation of
flows, and feasible compressor control, so pressure remains within limits (guaranteed)?
• Previous work:
– Objective function: minimize energy used by compressors
– Subject to known withdrawals of gas from the network and physical constraints
Andrzej Osiadacz (University of Warsaw)
Hans Aalto (Neste Jacobs)
Mohammad Abbaspour (Kansas State University / Kinder Morgan)
Richard Carter (Advantica / DNV-GL)
Marc Steinbach (Zuse Institute Berlin)
Modeling physics & engineering of gas pipelines for
large-scale control
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Challenges and approach
• What to consider in gas pipeline modeling
– Systems are large, distributed, complex, with many degrees of freedom
– Pressure, flow, and line pack changes propagate slowly; dynamics are highly nonlinear
– Boundary flows are always changing; flow never stabilizes to steady-state
– Thermal effects are highly localized (near compressors)
– Flow scheduling and compressor operations do not use optimization or model-based engineering
– Experience-based decisions and labor intensive control by human operators
• Our approach
– Consider basic components: pipes and compressors that connect nodes
– Model physical relationships (pressure, flow, compression)
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Modeling goals for transient optimization
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• Network nodes: physical nodes and custodial meter stations
• Network edges: pipes that connect nodes
• Compressors: machines that boost pressure
• Management objectives: operational or economic
– Operational: minimize cost of operations (energy use of compressors)
– Economic: maximize profit of gas delivery to buyers minus cost of gas supplied
• Conducted subject to engineering constraints on gas pipeline network
– Physics of pressure and flow on each pipe
– Flow balance at nodes
– Constraints on compressors
• Control parameters
– Compressor operations
– Nodal injections or withdrawals
Modeling for Transient Optimization
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Network Control
DynamicOptimization
Scalable Solver
(Sparse)
Algorithms
Computation
Modeling
Complex Fluid Dynamics
(PDEs)
Reduced Models(ODEs)
Physics
Network Science
Optimal Dynamic Compression Controls and Flow Schedule (solution)
Spatiotemporal Gas Withdrawal Constraints
(input)
Feasible System-wide Pressure(result)
High-fidelity simulation(validation)
Coarse-grained optimization(solution)
Physics on a pipe
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Isothermal Euler equations in one dimension:
• Mass conservation: 𝜕𝑡𝜌 + 𝜕𝑥 𝑢𝜌 = 0
• Momentum balance: 𝜕𝑡 𝜌𝑢 + 𝜕𝑥 𝜌𝑢2 + 𝜕𝑥𝑝 = −𝜆𝜌𝑢 𝑢
2𝐷− 𝜌𝑔sin(𝜃)
• State equation: 𝑝 = 𝑍𝑅𝑇𝜌
• 𝜌 ≡ density (kg/m3), 𝑝 ≡ pressure (Pa), 𝑢 ≡ velocity (m/s),
𝐷 ≡ diameter (m), 𝜆 ≡ friction factor, 𝜃 ≡ pipe angle (deg),
𝑍 ≡ gas compressibility factor, 𝑅 ≡ ideal gas constant (J/kg K),
𝑇 ≡ Temperature (K), 𝑔 ≡ velocity (m2/s),
Assume isothermal, simplified flow in a horizontal pipe without shocks:
• 𝑎 = 𝑍𝑅𝑇 is constant speed of sound (m/s),
• Neglect advection term 𝜕𝑥(𝜌𝑢2) and set 𝜃 = 0
• Define flow rate 𝜙 = 𝜌𝑢 (kg/m2/s)
Simplified equations: 𝜕𝑡𝜌 + 𝜕𝑥𝜙 = 0
𝜕𝑡𝜙 + 𝑎2𝜕𝑥𝜌 = −𝜆
2𝐷
𝜙 𝜙
𝜌
Reduced modeling of a pipeline segment
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Pipeline system model is an actuated PDE system on a metric graph:
• Set of nodes (junctions) 𝒱 and edges (pipes) ℰ
• Edges 𝑖, 𝑗 ∈ ℰ of length 𝐿𝑖𝑗, diameter 𝐷𝑖𝑗, and friction coefficient 𝜆𝑖𝑗
• Flow 𝜙𝑖𝑗 𝑡, 𝑥𝑖𝑗 and density 𝜌𝑖𝑗 𝑡, 𝑥𝑖𝑗 on an edge 𝑖, 𝑗 ∈ ℰ are continuous functions
of distance 𝑥 at all times 𝑡
Notations and definitions:
• Boundary densities 𝜌𝑖𝑗 𝑡 = 𝜌𝑖𝑗 𝑡, 0 and 𝜌𝑖𝑗𝑡 = 𝜌𝑖𝑗 𝑡, 𝐿
• Boundary flows 𝜙𝑖𝑗 𝑡 = 𝜙𝑖𝑗 𝑡, 0 and 𝜙𝑖𝑗𝑡 = 𝜙𝑖𝑗 𝑡, 𝐿
• Edges 𝑖, 𝑗 ∈ ℰ of length 𝐿𝑖𝑗, diameter 𝐷𝑖𝑗, and friction coefficient 𝜆𝑖𝑗
• Pressure nodes 𝑗 ∈ 𝒱𝑆 ⊂ 𝒱 with given density 𝑠𝑗 for 𝑗 ∈ 𝒱𝑆
• Flow nodes 𝑗 ∈ 𝒱𝐷 ⊂ 𝒱 with flow withdrawal (injection) 𝑑𝑗 for 𝑗 ∈ 𝑉𝐷
• Auxiliary nodal pressure variables 𝜌𝑗 for 𝑗 ∈ 𝒱𝐷
• Compressors for 𝑖, 𝑗 ∈ 𝒞 ⊂ ℰ modeled as boost from a node to pipe boundary:
𝜌𝑖𝑗 𝑡 = 𝛼𝑖𝑗𝜌𝑖 or 𝜌𝑖𝑗𝑡 = 𝛼𝑖𝑗𝜌𝑗 as appropriate
Weymouth equations in steady state:
• 𝜌𝑖𝑗2 − 𝜌
𝑖𝑗
2=
𝜆𝐿
𝐷𝑎2𝜙𝑖𝑗|𝜙𝑖𝑗| or 𝑝𝑖𝑗
2 − 𝑝𝑖𝑗
2= 𝛽𝑖𝑗𝜙𝑖𝑗|𝜙𝑖𝑗|, where 𝛽𝑖𝑗 =
𝜆𝐿𝑎2
𝐷
Flow balance (Kirchhoff-Neumann conditions):
𝑑𝑗 =
𝑖∈𝜕+𝑗
𝑋𝑖𝑗𝜙𝑖𝑗−
𝑖∈𝜕−𝑗
𝑋𝑖𝑗𝜙𝑖𝑗 ∀𝑗 ∈ 𝒱
Aggregate compressor stations to point objects:
Pressure boost by compressors:
𝜌𝑖𝑗 𝑡 = 𝛼𝑖𝑗𝜌𝑖 or 𝜌𝑖𝑗𝑡 = 𝛼𝑖𝑗𝜌𝑗 ∀(𝑖, 𝑗) ∈ ℰ
Modeling nodes and compressors
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Pipeline simulation: predictive analytics
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Initial Pressure Inlet Pressure Outlet Flow
Outlet Pressure Inlet Flow
Pipeline simulation
Inputs
• Initial conditions (pressure and flow)
• Either flow or pressure at each node over a
time interval T
Simulation
• Initial value problem with unique solution
Outputs
• Flows and pressures throughout the system
Simulation of catastrophic depressurization
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Boundary Conditions for Damage
• Change boundary condition at location node 𝑗 from
time 𝑡𝑑 of depressurization to 𝑝𝑗 𝑡 =𝑝atm for 𝑡 ≥ 𝑡𝑑
Boundary Conditions for Containment
• Set flow at upstream and downstream pipe
endpoints to 𝜙𝑖𝑗 𝑡 = 0 and 𝜙𝑗𝑘
𝑡 = 0 for 𝑡 ≥ 𝑡𝑐,
where 𝑡𝑐 = 𝑡𝑑 + 𝑡Δ is the valve closing time and 𝑡Δis the time elapsed until operators take action
• R. Hajossy, I. Mračka, P. Somora, and T. Žáčik, “Cooling of a wire as the model for a rupture location”, Mathematical
Institute, Slovak Academy of Sciences, In PSIG Annual Meeting. Pipeline Simulation Interest Group, 2014.
Pipeline flow in catastrophic depressurization
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Simulation of pressure, first 50 seconds after rupture
0
1000000
2000000
3000000
4000000
5000000
6000000
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
Pressure
sec -1 Pressure sec 0 Pressure sec 1 Pressure sec 2 Pressure
sec 3 Pressure sec 5 Pressure sec 6 Pressure sec 7 Pressure
sec 8 Pressure sec 9 Pressure sec 10 Pressure sec 11 Pressure
sec 20 Pressure sec 30 Pressure sec 40 Pressure sec 50 Pressure
Pre
ssu
re, P
a
Distance, m
A B
A B
-2000
-1500
-1000
-500
0
500
1000
1500
2000
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
Flow
sec -1 Mass flow sec 0 Mass flow sec 1 Mass flow sec 2 Mass flow
sec 3 Mass flow sec 5 Mass flow sec 6 Mass flow sec 7 Mass flow
sec 8 Mass flow sec 9 Mass flow sec 10 Mass flow sec 11 Mass flow
sec 20 Mass flow sec 30 Mass flow sec 40 Mass flow sec 50 Mass flow
Pipeline flow in catastrophic depressurization
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Simulation of mass flow, first 50 seconds after ruptureF
low
, kg/s
Distance, m
A B
A B
200
210
220
230
240
250
260
270
280
290
300
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
Temperature
sec -1 Temperature sec 0 Temperature sec 1 Temperaturesec 2 Temperature sec 3 Temperature sec 5 Temperaturesec 6 Temperature sec 7 Temperature sec 8 Temperaturesec 9 Temperature sec 10 Temperature sec 11 Temperaturesec 20 Temperature sec 30 Temperature sec 40 Temperaturesec 50 Temperature
Pipeline flow in catastrophic depressurization
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Simulation of temperature, first 50 seconds after rupture
Te
mp
era
ture
, d
egre
es K
Distance, m
A B
A B
Model validation using real data set
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• Reduced model of subsystem used for capacity
planning for a real pipeline
– 78 nodes, 91 pipes, 4 compressors 31 custody transfer
meters at 24 locations (labelled A to X)
– Hourly SCADA time-series of pressure and flow at
meters for a month during “polar vortex” conditions
– Model is validated in resolving hourly pressure
dynamics with less than 3% relative error
• Comparing relative distance (%) of SCADA
vs. simulation
– Pressure at flow nodes B to X
• Mean error: 4.17%
– Mass flow into system at node A
• Mean error: 2.45%
Pipeline subsystem model
• Meter stations
• Compressors
x distance (miles)
Pipeline diagram (not to scale)
• Zlotnik, Anatoly V., Aleksandr M. Rudkevich, Evgeniy Goldis, Pablo A. Ruiz, Michael Caramanis, Richard G. Carter, Scott N.
Backhaus, Richard Tabors, and Daniel Baldwin. “Economic optimization of intra-day gas pipeline flow schedules using
transient flow models.” in Proc. Pipeline Simulation Interest Group Annual Conf., 1715, Atlanta, GA, May 2017.
Transient optimization: decision analytics
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Inputs
• Desired outlet flow
• Objective (minimize compressor power)
Optimization
• A decision among many possibilities for the
best solution
Outputs
• Control of pressure by compressors
Results
• Guarantee feasibility for inequality constraints
• Optimal solution
Feasible Outlet Pressure Feasible Inlet Flow
Validation Simulation
Transient Optimization
Initial Pressure Outlet FlowControl: Inlet Pressure
Modeling gas pipelines for control
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Kg/
s
bo
ost
rat
io
Pressure trajectories
Flow trajectories
Compression Controls (solution)
time (24 hours)
Example system • 5 compressors, 8 loads, 1 source• 300 miles of pipes
Load profiles
• Zlotnik, Anatoly, Michael Chertkov, and Scott Backhaus. "Optimal control of transient flow in natural gas networks." In
Decision and Control (CDC), 2015 IEEE 54th Annual Conference on, pp. 4563-4570. IEEE, 2015.
• Model-predictive
optimal control of gas
pipelines
– Old paradigm: Given
predicted flow profiles,
how to operate
compressors such that
pressure remains within
set limits (if possible)?
Pipeline as conservation laws on directed metric graph
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Approximation with Nodal Boundary Conditions
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Graph representation of flow balance
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Graph representation of density gradients
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Matrix Differential Algebraic Equations
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Matrix Differential Algebraic Equations
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Matrix Differential Algebraic Equations
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Comparing discretization schemes
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Alternative discretizations• In space: trapezoidal rule (TZ)
and lumped elements (LU) • In time: pseudospectral
approximation (PS) and trapezoidal rule (TZ)
• Tested by two-stage scheme• First stage minimizes
compressor energy
• Second stage minimizes solution variation
• Mak, Terrence W. K., Hentenryck, P. V., Zlotnik, A., & Bent, R. (2019). Dynamic compressor optimization in natural gas
pipeline systems. INFORMS Journal on Computing, 31(1), 40-65.
Comparing discretization schemes
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TZ-TZ LU-TZ
TZ-PS LU-PS
Space-Time Schemes
Lumped Elements + Trap• Fastest and most accurate• Used in implementations
• Mak, Terrence W. K., Hentenryck, P. V., Zlotnik, A., & Bent, R. (2019). Dynamic compressor optimization in natural gas
pipeline systems. INFORMS Journal on Computing, 31(1), 40-65.
Gas market design using pipeline optimization
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Inputs and outputs
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• Input: static network model
– Junctions (nodes)
– pipes (edges)
– compressor stations (controllers)
– custody transfer meters (at nodes)
• Input: hourly bid from each shipper
– Pre-existing (ratable) flow schedule
– Bid or offer prices
– Upper limits on gas injections and
withdrawals at each price level (hourly)
• Output: physical solution
– Pressures and flows through the pipeline
– Compressor controls (discharge pressure)
– Validated in simulation, to control room
• Output: market solution
– Locational trade values (LTVs)
give real-time and forward prices
– Flow profiles of increment or decrease
w.r.t. ratable nomination (private to each
shipper)
Constraints on compressors:
• Maximum compressor power: 𝜂𝑖𝑗 𝜙𝑖𝑗 𝛼𝑖𝑗2𝑚 − 1 ≤ 𝐸𝑖𝑗
max
∀ 𝑖, 𝑗 ∈ 𝒞• Minimum boost ratio: 𝛼𝑖𝑗 ≥ 1 ∀ 𝑖, 𝑗 ∈ 𝒞
Constraints on pipe pressure:
• Minimum pressure: 𝑝𝑗 ≥ 𝑝𝑗min ∀𝑗 ∈ 𝒱
• Maximum pressure: 𝛼𝑖𝑗𝑝𝑗 ≤ 𝑝𝑖𝑗max ∀ 𝑖, 𝑗 ∈ ℰ
Objective:
• Reflect goals of pipeline system manager and security priorities
• Maximize economic value (social welfare) produced by the system
• Minimize energy cost (maximize efficiency) of running the system
• Prioritize critical assets
𝐽𝑀𝑆𝑊 =
𝑘∈𝒱
𝑐𝑘𝑜𝑑𝑘 −
𝑘∈𝒱
𝑐𝑘𝑠𝑠𝑘 −
𝑖,𝑗 ∈𝒞
𝜂𝑖𝑗 𝜙𝑖𝑗 𝛼𝑖𝑗2𝑚 − 1
Constraints and objective
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Economic Optimal Control Formulation
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Economic Optimal Control Formulation
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Objective: Social (Economic)
Welfare or Market surplus
(may be regularized by adding
Cost of compressor operation)
Economic Optimal Control Formulation
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Constraints and
Lagrange multipliers
Economic Optimal Control Formulation
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Equality (dynamic)
constraints: Partial Differential
Equations for gas flow dynamics
in terms of pressure and flow
Economic Optimal Control Formulation
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Equality constraints: Mass flow
balance at nodes separated into
baseline and optimized demand
and supply flows at transfer points
Economic Optimal Control Formulation
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Equality constraints: Pressure
boost of compressors, modeled as
relation between pressure
(density) at a node and at the pipe
boundary
Economic Optimal Control Formulation
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Inequality constraints:
Operating limits on pressure
inside each pipe (specifically,
upper bound in pipe, and lower
bound at nodes to guarantee
contractual pressure)
Economic Optimal Control Formulation
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Inequality constraints:
Operating limits on compressors
Specifically, upper bound on
applied power, and lower bound of
unity boost ratio (bypassed if off)
Economic Optimal Control Formulation
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Inequality constraints: Injection or
withdrawal to/from the system,
upper and lower bounds on supply
(injection) and demand (withdrawal)
Economic Optimal Control Formulation
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Participant bids: time-dependent
price and quantity bids of buyers
and sellers
Economic Optimal Control Formulation
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Locational Trade Values: time-
dependent nodal prices of gas and
spatiotemporal prices of momentum
(transportation) and mass (gas in
the pipe)
Time-periodic Formulation
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Well-posedness: time-periodicity
constraint for well-posed problem,
represents mass-preservation
Computational example
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Pipeline test network: 24 pipes, 5
compressors, 477 km
Input data: baseline flows and
price/quantity bids
Computational example
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Output: physical and price
solutions
State and parameter estimation
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Representing uncertainty for estimation model
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• Account for uncertainty using a noise process η
– simplification of physical modeling
– Uncertainty in model parameters
– process and measurement noise
• Minimize estimation error using least squares objective
State estimation problem
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Estimation for synthetic data
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Good model identification
for synthetic data:
• Kaarthik Sundar and Anatoly Zlotnik. “State and Parameter Estimation for Natural Gas Pipeline Networks Using Transient
State Data.” in IEEE Transactions on Control Systems Technology, 27:5, 2110 – 2124, 2019.
Joint state and parameter estimation for real data
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Pipeline subsystem model
• Meter stations
• Compressors
x distance (miles)
• Reduced model of subsystem used for capacity
planning for a real pipeline
– 78 nodes, 91 pipes, 4 compressors 31 custody transfer
meters at 24 locations (labelled A to X)
– Hourly SCADA time-series of pressure and flow at
meters for a month during congested conditions
Metered pressures (Mpa)Friction factor estimates
Monotonicity properties and modeling implications
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Physical flow network as a directed metric graph
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Physical flow network as a directed metric graph
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Distributed dynamics on edges
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Nodal compatibility conditions
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Well-posedness & regularity assumptions
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Well-posedness & regularity assumptions
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Theorem: monotone order propagation
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Application: monotone parameterized control systems
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Application: robust optimal control
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Application: friction-dominated models
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• Friction dominated modeling
– Omit flux derivative term, approximate hyperbolic system by parabolic one
– Proposed to simplify mathematical modeling for simulation and optimization
– Herty, Michael, Jan Mohring, and V. Sachers. "A new model for gas flow in pipe
networks." Mathematical Methods in the Applied Sciences 33, no. 7 (2010): 845-855.
• Monotonicity property for gas pipelines assumes friction-dominated flow
– Under what conditions are these assumptions valid?
𝜕𝑡𝜌 + 𝜕𝑥𝜙 = 0
𝜕𝑡𝜙 + 𝑎2𝜕𝑥𝜌 = −𝜆
2𝐷
𝜙 𝜙
𝜌
𝜕𝑡𝜌 + 𝜕𝑥𝜙 = 0
𝑎2𝜕𝑥𝜌 = −𝜆
2𝐷
𝜙 𝜙
𝜌
𝑎2𝜕𝑥𝜌 = −𝜆
2𝐷
𝜙 𝜙
𝜌
• Left: Fast Transients
– Flow in a single pipe with sinusoidal
variation (3 cycles over 1 hour) in outlet
flow with maximum magnitudes of 120,
300, 400, and 600 kg/s.
– The monotonicity theorem does not
apply for the fast transient regime
• Right: Slow Transients
– Flow in a single pipe with slow sinusoidal
variation (3 cycles in 24 hours) in outlet
flow with max magnitudes of 120, 300,
400, & 600 kg/s.
– The monotonicity theorem holds in the
slow transient regime
• Guidance: friction-dominated
modeling should not be used to
represent fast transients
Application: friction-dominated models
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• Misra, Sidhant, Marc Vuffray, and Anatoly Zlotnik. "Monotonicity Properties of Physical Network Flows and Application to
Robust Optimal Allocation." Proceedings of the IEEE (to appear), arXiv:2007.10271.
• Testing the monotonicity property in the normal operating regime of gas pipelines
– Top left: Baseline withdrawals (kg/s) custody transfer stations.
– Top right: Increase of withdrawals above baseline by 5%.
– Bottom left: Simulated pressure (PSI) solutions given baseline withdrawals.
– Bottom right: Simulated pressure solutions given increased withdrawals.
• Monotonicity property can be invoked in practice for transient optimization
Application: real data validation
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• Misra, Sidhant, Marc Vuffray, and Anatoly Zlotnik. "Monotonicity Properties of Physical Network Flows and Application to
Robust Optimal Allocation." Proceedings of the IEEE (to appear), arXiv:2007.10271.
Coordination of electricity and gas transmission
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Initial study on benefits of joint optimization
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• Joint optimization problem
– Simple model of real-time load balancing by a optimal power flow
– Combined with transient optimization of gas flows
• Coordination scenarios
– 1: Status quo systems and markets
– 2: Predictive dynamic gas flow control
– 3: Joint optimization with status quo methods (steady-state\static gas system settings)
– 4: Joint optimization with dynamic gas flow control
• System stress cases
– Base case (regular operations)
– Stress case (systems at capacity)
• Zlotnik, Anatoly, Line Roald, Scott Backhaus, Michael Chertkov, and Göran Andersson. "Coordinated scheduling for
interdependent electric power and natural gas infrastructures." IEEE Transactions on Power Systems 32, no. 1 (2017): 600-610.
Initial study on benefits of joint optimization
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• Joint optimization problem
– Simple model of
real-time load
balancing by OPF
– Transient optimization
of gas flows
– Coupled through
heat rate curve of
gas-fired generators
• Zlotnik, Anatoly, Line Roald, Scott Backhaus, Michael Chertkov, and Göran Andersson. "Coordinated scheduling for
interdependent electric power and natural gas infrastructures." IEEE Transactions on Power Systems 32, no. 1 (2017): 600-610.
Initial study on benefits of co-optimization
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Power system model
Dynamic constraints
on gas availability
Gas pipeline network model
• Simple model
– Fixed gas price $/mmBTU,
– Quadratic heat rate curves,
– Quadratic generation cost curves
Initial study on benefits of co-optimization
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Initial study on benefits of co-optimization
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Initial study on benefits of co-optimization
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A realistic coordination mechanism
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• Realistic coordination between sectors
– A gas balancing market using rolling horizon model-predictive optimal control
– Fits into decision cycles for day-ahead scheduling of electricity and gas systems
Evaluating the coordination concept
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• Optimization model for power system
– Standard Unit Commitment (UC) for day-ahead market
– Mixed Integer Linear Program, control variables are generator production
– Objective function is minimum production cost
– Constraints on power system and generators
• Optimization model for gas system
– Optimal control of flows on a network, control variables are compressors & demands
– Objective function is maximizing economic welfare for system users
– Dynamic constraints are PDEs on network edges, Kirchoff’s law on nodes
– Inequality constraints on states and controls
• Iterative coordination mechanism between two models
– Limited to exchange of generation/flow and price time-series (not network models)
Evaluating the coordination concept
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Power
System
(Unit
Commitment)
Gas
System
(Gas
Balancing
Market)
Generator
(Heat
Rate
Curve)
• A study to test the mechanism:
Evaluating the coordination concept
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Power
System
(Unit
Commitment)
Gas
System
(Gas
Balancing
Market)
Generator
(Heat
Rate
Curve)
Optimal Production
Schedule 𝑝𝑖(𝑡)
Locational Marginal
Prices 𝜆𝑖𝑝(𝑡)
• A study to test the mechanism:
𝑑𝑖max(𝑡): Maximum gas
demand of generators
Bid (buy) price 𝑐𝑖𝑔(𝑡)
for gas
𝑑𝑖max = ℎ1(𝑝𝑖)
𝑐𝑖𝑔= ℎ2(𝜆𝑖
𝑝)
Evaluating the coordination concept
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Power
System
(Unit
Commitment)
Gas
System
(Gas
Balancing
Market)
Generator
(Heat
Rate
Curve)
Optimal Production
Schedule 𝑝𝑖(𝑡)
Locational Marginal
Prices 𝜆𝑖𝑝(𝑡)
• A study to test the mechanism:
𝑑𝑖max(𝑡): Maximum gas
demand of generators
Bid (buy) price 𝑐𝑖𝑔(𝑡)
for gas
𝑝𝑖max(𝑡): Maximum
Production Schedule
Optimal gas delivery
to power generators
𝑑𝑖(𝑡) ≤ 𝑑𝑖max(𝑡)
Locational Trade
Values of gas 𝜆𝑖𝑔(𝑡)
𝑐𝑖𝑝(𝑡): Marginal price
of generation (of fuel)
𝑑𝑖max = ℎ1(𝑝𝑖)
𝑐𝑖𝑔= ℎ2(𝜆𝑖
𝑝)
𝑝𝑖 = ℎ1−1(𝑑𝑖
max)
𝑐𝑖𝑝= ℎ2
−1(𝜆𝑖𝑔)
• Zhao, Bining, Anatoly Zlotnik, Antonio J. Conejo, Ramteen Sioshansi, and Aleksandr M. Rudkevich. "Shadow Price-Based
Coordination of Natural Gas and Electric Power Systems." IEEE Transactions on Power Systems (2018).
Computational Example
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24 pipe gas test network 24 node IEEE RTS power network System power demand profile
• Procedure converges after 1
iteration!
Computational Example
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Generation Schedule:
1 hour increments
Generation Schedule:
15 minute increments
Hourly electricity price
Initial Iteration
Final iteration
• Gas generators have lowest
marginal costs
• Production is transferred to
other sources
Path to gas-electric system interoperability
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• Gas-electric coordination using optimization-based markets
– Time-dependent locational marginal pricing (electricity LMPs and natural gas LTVs)
– Requires only limited exchange of information to produce price/quantity (P/Q) bids and
production/demand constraints
• Properties
– Revenue adequacy for the administrators of both markets
– Operation of systems is not altered if all demands can be met
– Convergence after only one iteration of the procedure (by ~linearity of UC)
• Zhao, Bining, Anatoly Zlotnik, Antonio J. Conejo, Ramteen Sioshansi, and Aleksandr M. Rudkevich. "Shadow Price-Based
Coordination of Natural Gas and Electric Power Systems." IEEE Transactions on Power Systems (2018).
Transition to practice
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Commercial ENELYTIX system
Power System Optimizer (PSO) by Polaris (CPLEX).
Gas System Optimizer (GSO) by LANL (IPOPT).
Scalable and flexible cloud-based architecture.
Coordinated Operation of Electric And Natural Gas Supply
Networks: Optimization Processes And Market Design
• Zlotnik, Anatoly, Sundar, Kaarthik, Rudkevich, Alexandr. M., Tabors, Richard, & Li, Xindi. "Pipeline Transient Optimization for
a Gas-Electric Coordination Decision Support System." PSIG Annual Meeting. Pipeline Simulation Interest Group, 2019.
Fuel Reliability for Electric Energy Delivery by Optimized
Management of Gas-pipeline Automation Systems (FREEDOM GAS)
Software development, system integration, and pilot study
Acknowledgement
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• ARPA-e Project GECO
– Advanced Research Project Agency-Energy (ARPA-e) of the
U.S. Department of Energy, Award No. DE-AR0000673
• Kinder Morgan, PJM
• Advanced Grid Modeling Program
– D.O.E. Office of Electricity
– D.O.E. Office of Energy Efficiency and Renewable Energy
• Los Alamos National Laboratory
– National Nuclear Security Administration
of the U.S. Department of Energy
under Contract No. 89233218CNA000001
• GRAIL: Gas Reliability Analysis Integrated Library
– Open source LANL-developed prototype algorithms
https://github.com/lanl-ansi/grail (OSTI ID 18546)
Questions?