Practical Mathematical Modeling for Simulation, Estimation ... · 7/22/2020 · Practical Mathematical Modeling for Simulation, Estimation, and Optimal Control of Gas Pipeline Systems

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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

https://lanl-ansi.github.io/

Collaboration


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


• 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


Grid modernization


• 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


• 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


• 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


• 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


• 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


Energy systems are now coupled


• 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


• 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


Power grid basics


• 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


$800

$6

PJM Interconnection price per MWh

July 19, 2013 heat wave

Motivation


• 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


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)


Modeling goals for transient optimization


• 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


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


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


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


Pipeline simulation: predictive analytics


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


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


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



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





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



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


• 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


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


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


Approximation with Nodal Boundary Conditions


Graph representation of flow balance


Graph representation of density gradients


Matrix Differential Algebraic Equations






Comparing discretization schemes


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


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


Inputs and outputs


• 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


Economic Optimal Control Formulation




Objective: Social (Economic)

Welfare or Market surplus

(may be regularized by adding

Cost of compressor operation)



Constraints and

Lagrange multipliers



Equality (dynamic)

constraints: Partial Differential

Equations for gas flow dynamics

in terms of pressure and flow



Equality constraints: Mass flow

balance at nodes separated into

baseline and optimized demand

and supply flows at transfer points



Equality constraints: Pressure

boost of compressors, modeled as

relation between pressure

(density) at a node and at the pipe

boundary



Inequality constraints:

Operating limits on pressure

inside each pipe (specifically,

upper bound in pipe, and lower

bound at nodes to guarantee

contractual pressure)



Inequality constraints:

Operating limits on compressors

Specifically, upper bound on

applied power, and lower bound of

unity boost ratio (bypassed if off)



Inequality constraints: Injection or

withdrawal to/from the system,

upper and lower bounds on supply

(injection) and demand (withdrawal)



Participant bids: time-dependent

price and quantity bids of buyers

and sellers



Locational Trade Values: time-

dependent nodal prices of gas and

spatiotemporal prices of momentum

(transportation) and mass (gas in

the pipe)

Time-periodic Formulation


Well-posedness: time-periodicity

constraint for well-posed problem,

represents mass-preservation

Computational example


Pipeline test network: 24 pipes, 5

compressors, 477 km

Input data: baseline flows and

price/quantity bids

Computational example


Output: physical and price

solutions

State and parameter estimation


Representing uncertainty for estimation model


• 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


Estimation for synthetic data


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


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


Physical flow network as a directed metric graph


Physical flow network as a directed metric graph


Distributed dynamics on edges


Nodal compatibility conditions


Well-posedness & regularity assumptions


Well-posedness & regularity assumptions


Theorem: monotone order propagation


Application: monotone parameterized control systems


Application: robust optimal control


Application: friction-dominated models


• 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


• 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


• 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


Initial study on benefits of joint optimization


• 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


• 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


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







A realistic coordination mechanism


• 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


• 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)



Power

System

(Unit

Commitment)

Gas

System

(Gas

Balancing

Market)

Generator

(Heat

Rate

Curve)

• A study to test the mechanism:



Power

System

(Unit

Commitment)

Gas

System

(Gas

Balancing

Market)

Generator

(Heat

Rate

Curve)

Optimal Production

Schedule 𝑝𝑖(𝑡)

Locational Marginal

Prices 𝜆𝑖𝑝(𝑡)


𝑑𝑖max(𝑡): Maximum gas

demand of generators

Bid (buy) price 𝑐𝑖𝑔(𝑡)

for gas

𝑑𝑖max = ℎ1(𝑝𝑖)

𝑐𝑖𝑔= ℎ2(𝜆𝑖

𝑝)



Power

System

(Unit

Commitment)

Gas

System

(Gas

Balancing

Market)

Generator

(Heat

Rate

Curve)

Optimal Production

Schedule 𝑝𝑖(𝑡)

Locational Marginal

Prices 𝜆𝑖𝑝(𝑡)


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


24 pipe gas test network 24 node IEEE RTS power network System power demand profile

• Procedure converges after 1

iteration!

Computational Example


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


• 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


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


• 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)

https://github.com/lanl-ansi/grail

Questions?

[email protected]

Practical Mathematical Modeling for Simulation, Estimation ... · 7/22/2020 · Practical Mathematical Modeling for Simulation, Estimation, and Optimal Control of Gas Pipeline Systems

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