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An Integrated Market for Electricity and Natural Gas Systems with Stochastic Power Producers

Christos Ordoudis, Pierre Pinson, Juan M. Morales

arXiv:1805.04414v2math.OC

TL;DR

Imperfect coordination between electricity and natural gas markets, alongside uncertain power supply, motivates integrated operational modeling. The paper proposes stochastic co-optimization with gas linepack, and reports that full coupling copes with power-supply uncertainty while making use of natural-gas flexibility.

  • Problem

    The paper addresses coordination of electricity and natural gas systems under uncertain power supply.

    Source: §1 · p. 2

  • Method

    The paper proposes a stochastic co-optimization model for integrated electricity and natural gas systems that includes natural-gas linepack in a MILP formulation.

    Source: §2.2.3 · p. 8, §6 · p. 25

  • Results

    The fully coupled model copes with power-supply uncertainty and makes use of natural-gas system flexibility, increasing operational flexibility.

    Source: §6 · pp. 25–26

  • Takeaways & Limitations

    The conclusion identifies natural-gas linepack modeling as a way to capture flexibility relevant to integrated system operation.

    Source: §6 · p. 25

    Future work includes optimizing coordination-parameter definitions to reduce expected system cost and providing a more detailed natural-gas description.

    Source: §6 · p. 26

Abstract

from arXiv · show

In energy systems with high shares of weather-driven renewable power sources, gas-fired power plants can serve as a back-up technology to ensure security of supply and provide short-term flexibility. Therefore, a tighter coordination between electricity and natural gas networks is foreseen. In this work, we examine different levels of coordination in terms of system integration and time coupling of trading floors. We propose an integrated operational model for electricity and natural gas systems under uncertain power supply by applying two-stage stochastic programming. This formulation co-optimizes day-ahead and real-time dispatch of both energy systems and aims at minimizing the total expected cost. Additionally, two deterministic models, one of an integrated energy system and one that treats the two systems independently, are presented. We utilize a formulation that considers the linepack of the natural gas system, while it results in a tractable mixed-integer linear programming (MILP) model. Our analysis demonstrates the effectiveness of the proposed model in accommodating high shares of renewables and the importance of proper natural gas system modeling in short-term operations to reveal valuable flexibility of the natural gas system. Moreover, we identify the coordination parameters between the two markets and show their impact on the system's operation and dispatch.

1. Introduction

The introduction frames imperfect coordination between electricity and natural gas markets—and between trading stages—as a barrier to efficient operation with variable, uncertain renewables. It highlights natural-gas network flexibility as relevant to reliability, while noting limitations in prior short-term models.

1. Introduction

  • Static coordination parameters and imperfectly coordinated markets and trading floors limit efficient interaction between electricity and natural gas systems.The introduction also notes that sequential day-ahead and balancing auctions prevent intertemporal coordination that could facilitate integration of variable, uncertain renewables.Source: §1 · p. 2
  • Prior studies using steady-state natural-gas models may produce suboptimal solutions in short-term operations.The introduction points to linepack—the ability to store natural gas in pipelines—as a relevant dynamic feature.Source: §1 · pp. 2–3
  • Natural-gas storage in pipelines and slower movement give the gas system flexibility that can support reliability during variable gas withdrawals by gas-fired power plants.The introduction links this flexibility particularly to cases with highly variable GFPP withdrawals.Source: §1 · pp. 2–3
  • Earlier nonlinear linepack approaches do not guarantee global optima, can require high computation times, and make market pricing difficult.The introduction contrasts these approaches with a mixed-integer linear approximation of dynamic gas operation.Source: §1 · p. 3
  • Some prior integrated models impose natural-gas network constraints only for feasibility, without optimizing gas-system operational cost.The introduction says this means optimal operation of the integrated energy system is not guaranteed.Source: §1 · p. 3

2. Coordination Framework

This section develops a coordination framework for electricity and natural gas markets, examining system integration and trading-floor timing through market-clearing models. It compares independent or sequential operation with coupled stochastic co-optimization, including different treatments of gas-fired plants, renewable uncertainty, and balancing.

2.1. Market designs

  • Coupled markets account for gas-market prices and availability in power-plant dispatch, while asynchronous trading and lower-priority gas demand can constrain gas-fired plants.The proposed coupled market-clearing models contrast with a decoupled model resembling current designs with limited coordination.Source: §2.1 · pp. 5–6
  • Stochastic programming co-optimizes day-ahead and balancing dispatch, reducing operating costs when uncertain power production is represented by scenarios.This enables temporal coordination across trading floors.Source: §2.1 · p. 6

2.2. Market-clearing models

  • Three market-clearing models cover different levels of electricity–gas system and temporal coordination.Figure 2 depicts the model set and its day-ahead, balancing, electricity, and natural gas market dimensions.Source: §2.2 · p. 7, Figure 2 · p. 7

2.2.1. Model Seq-Dec – Sequential decoupled electricity and natural gas model

  • The sequential decoupled model treats electricity and natural gas operations independently and represents renewable production deterministically.Source: §2.2.1 · p. 7
  • Seq-Dec schedules electricity and natural gas independently, clearing electricity first and submitting gas-fired power plant fuel use as fixed gas demand.It bids gas-fired generation using an estimated spot or contract price, although actual gas value is represented by the locational marginal price at each system location.Source: §2.2.1 · p. 7
  • Residential gas loads have priority over gas-fired power plants, whose fuel use is iteratively constrained when schedules are infeasible.The independent approach can produce suboptimal or infeasible schedules under highly increased residential electricity and gas demands.Source: §2.2.1 · p. 7

2.2.2. Model Seq-Coup – Sequential coupled electricity and natural gas model

  • Seq-Coup jointly optimizes the two systems while day-ahead and balancing markets still clear sequentially, with gas-fired plant demand variable and production cost determined by local gas value.Renewable production is deterministic; the model is intended to highlight inefficiencies from independently scheduling the systems.Source: §2.2.2 · pp. 7–8

2.2.3. Model Stoch-Coup – Stochastic coupled electricity and natural gas model

  • Stoch-Coup co-optimizes the systems with uncertain renewable production in a two-stage stochastic model, achieving full integration and perfect temporal coordination.It allows pre-positioning while accounting for future balancing costs, unlike the deterministic models.Source: §2.2.3 · p. 8
  • The study compares coupled models with Seq-Dec to assess how system and temporal coordination affect expected cost and unit dispatch; it omits a decoupled stochastic model.That omitted alternative would retain separate system operation while using two-stage stochastic programming for temporal coupling of trading floors.Source: §2.2.3 · p. 8

3. Natural Gas Network Modeling

The model uses simplifying assumptions to represent key gas-network dynamics—including pipeline flows, compressors, linepack, and gas storage—in day-ahead and real-time market clearing. It linearizes gas-flow constraints and models storage explicitly, while noting that transient modeling would be more realistic but computationally more complex.

3. Natural Gas Network Modeling

  • The gas-network model incorporates flow characteristics, compressor branches, linepack and gas storage as day-ahead and real-time market-clearing decisions.The formulation uses simplifying assumptions to capture these dynamics; gas transport is assumed isothermal in horizontal pipelines.Source: §3 · p. 8, §3.1 · p. 9

3.1. Nodal and natural gas flow constraints

  • Nonlinear Weymouth flow constraints are approximated with Taylor-based outer-approximation inequalities at fixed pressure points, using planes tangent to the Weymouth cone.Fixed points are selected within adjacent-node pressure limits.Source: §3.1 · p. 9
  • Direction-specific pressure limits yield separate flow approximations, and a binary variable selects the applicable linearization for bidirectional pipelines.The model introduces separate non-negative variables for flow in each direction.Source: §3.1 · pp. 9–10

3.2. Natural gas storage constraints

  • Linepack is represented through pipeline gas mass proportional to adjacent-node pressures and a mass-conservation constraint that allows inflow and outflow to differ.The model defines pipeline flow as the average of inflow and outflow.Source: §3.2 · p. 11, §3.1 · p. 11
  • Gas-storage constraints impose temporal volume bounds based on capacity and cushion gas, as well as maximum inflow and outflow rates.Cushion gas is the minimum gas quantity needed to operate a storage unit.Source: §3.2 · p. 12

4. Market-clearing Models

This section presents market-clearing models for electricity and natural gas under wind uncertainty and network constraints, ranging from sequential decoupled operation to coordinated dispatch. It also covers stochastic scheduling that anticipates balancing needs.

4. Market-clearing Models

  • Wind uncertainty is represented by correlated scenarios, with gas-demand uncertainty arising from gas-fired power plants and inelastic demand supporting cost minimization.Load shedding is penalized, and the operator minimizes expected system cost.Source: §4 · p. 12
  • Linear production costs and competitive full-capacity offers yield MILP market-clearing formulations; fixing gas-flow direction binaries produces LPs whose balancing-constraint duals give prices.Wind producers bid at zero marginal cost.Source: §4 · pp. 12–13
  • The models impose electricity and gas network constraints over a multi-period horizon; coupled markets clear concurrently in day-ahead and balancing stages, without intraday trading.Electricity transmission uses DC power flow, and the same day-ahead/balancing timing is extended to natural gas for a consistent market design.Source: §4 · p. 12

4.1. Sequential decoupled electricity and natural gas model

  • The sequential-decoupled model clears electricity and gas independently, scheduling electricity day-ahead first and using its GFPP fuel consumption as time-varying gas demand.The electricity schedule uses expected wind production; the resulting GFPP demand is then treated as fixed in the gas day-ahead model.Source: §4.1 · pp. 13–14
  • After wind realization, the model fixes day-ahead schedules and clears electricity and gas balancing independently, passing GFPP fuel-consumption deviations into gas demand.Gas linepack provides a practically free first redispatch response; costly gas redispatch is activated when linepack flexibility is unavailable.Source: §4.1 · pp. 14–16
  • When GFPP fuel use causes residential gas load shedding, the decoupled procedure constrains affected plants’ balancing output and repeats until the shedding stops.The procedure identifies the responsible GFPPs and system nodes before imposing the production bounds.Source: §4.1 · p. 16

4.2. Sequential coupled electricity and natural gas model

  • Day-ahead and balancing markets clear sequentially, but electricity and gas operations are coordinated, with GFPP fuel consumption variable and its costs assigned to the gas system.Excluding GFPP power-production costs from the electricity objective avoids double counting because fuel costs enter through the gas-system balance.Source: §4.2 · p. 17

4.3. Stochastic coupled electricity and natural gas model

  • The stochastic coupled model coordinates day-ahead and balancing decisions implicitly by minimizing expected balancing costs over wind scenarios and anticipating future balancing needs.Day-ahead wind production is limited by installed capacity.Source: §4.3 · p. 18
  • Flexible producers may be scheduled out of merit order to provide balancing flexibility, while gas constraints enable spatial allocation of linepack flexibility near real time.The model includes natural-gas constraints in both day-ahead and balancing stages.Source: §4.3 · pp. 18–19

5. Results

The results assess how coordinated stochastic scheduling and natural-gas linepack modeling shape system costs, dispatch, and gas-network flexibility under uncertain wind. They also examine how these outcomes depend on gas-price assumptions, initial linepack, residential demand, and scenario and real-time regulation needs.

5.1. Comparison of market-clearing models and the effect of coordination parameters

  • Gas-price misestimation raises day-ahead costs by misplacing GFPPs in the merit order and scheduling more expensive units.The comparison is against Seq-Dec using the actual natural gas price from Seq-Coup; the simulated estimation error is 10% over or under.Source: §5.1 · pp. 20–21
  • Across the two reported wind-penetration conditions—40% and 50%—Stoch-Coup achieves the lowest expected cost in both cases, according to the accompanying expected-cost table captions.Source: §5.1 · p. 20, Table 1 · p. 21, Table 2 · p. 21
  • Stoch-Coup accounts for wind uncertainty with higher day-ahead but lower expected balancing costs, and increases the share of efficient balancing GFPPs relative to deterministic models.The deterministic models schedule by merit order, which may not be suitable for wind uncertainty.Source: §5.1 · p. 20
  • Coupled clearing lowers expected cost versus decoupled approaches by optimizing natural gas price signals, flows, and linepack, whose spatial allocation affects short-term adequacy and flexibility.The passage attributes this benefit to gas prices and network operation being jointly coordinated.Source: §5.1 · p. 20
  • When natural gas price is over-estimated, total expected cost can be reduced; at 50% wind penetration, flexible GFPPs are scheduled less at the day-ahead stage, making them available for real-time upward regulation.Source: §5.1 · p. 21
  • During peak residential natural gas load, GFPP 11 is expected to face unsatisfied fuel demand under Seq-Dec, but not Seq-Coup, highlighting efficient gas-flow optimization and spatial linepack allocation.Source: §5.1 · p. 21

5.2. The benefits and effects of linepack

  • Neglecting linepack increases day-ahead costs in Seq-Coup and Seq-Dec but lowers expected balancing costs compared with models that include linepack.Without linepack, demand must be met at each time period; with it, gas can be stored in the network and used later during peak demand.Source: §5.2 · p. 22
  • Linepack makes cheaper natural-gas down-regulation more available, which supports GFPPs and lowers expected electricity balancing cost, depending on scenarios and real-time regulation needs.The passages characterize this as a possible source of inefficiency when linepack flexibility is not properly represented.Source: §5.2 · pp. 22–23, Table 4 · p. 23
  • With linepack, Stoch-Coup has lower expected cost than the steady-state model and schedules more gas-fired power plants (GFPPs) by exploiting gas-system storage.The passages identify deterministic approaches as myopic to uncertainty and say temporal coordination allocates natural gas efficiently.Source: §5.2 · p. 23, Table 5 · p. 23
  • The performance ratio shows that modeling natural-gas-network linepack flexibility unveils 12.4% of the flexibility available from utilizing an ideal storage facility in the electricity system.In this case, Stoch-Coup’s expected cost (EC) is $1 629 519.Source: §5.2 · p. 23
  • Higher initial linepack lowers total operating cost by using free energy stored in the gas network and results in scheduling more GFPPs than Stoch-Coup; expected cost is higher when gas must be filled to meet the horizon’s final condition.Source: §5.2 · p. 24
  • Linepack use and gas production respond to residential demand: the system draws down storage during peak demand to avoid expensive gas supply, and GFPP fuel use flexes with gas availability.When initial linepack is higher, gas production falls early and GFPPs are scheduled more; GFPPs thus act as flexible natural-gas demand.Source: §5.2 · pp. 24–25

6. Conclusion

The paper proposes an integrated electricity–natural gas market-clearing model that co-optimizes under uncertain power supply and represents pipeline linepack in a tractable MILP. Its analysis finds that coordination, parameter choices, and appropriate gas-system modeling affect operational flexibility and expected system cost.

6. Conclusion

  • The proposed model co-optimizes electricity and natural-gas systems, approximating gas-system dynamics in a tractable mixed-integer linear program that includes pipeline linepack.Linepack represents the possibility of storing gas in network pipelines, which is important in short-term operations.Source: §6 · p. 25
  • Combining the model properties increases operational flexibility and improves gas-resource allocation in the network.The conclusion also reports that simulations capture natural-gas-system behavior efficiently.Source: §6 · pp. 25–26
  • The paper provides deterministic market-clearing models alongside its proposed approach to assess coordination between energy systems and trading floors.The deterministic models use a deterministic description of uncertain power supply.Source: §6 · p. 25
  • Gas-fired power-plant consumption and natural-gas price affect both systems’ operation and expected cost, and poor parameter definitions usually increase expected cost.Distorted day-ahead scheduling can make reserve capacity available that suits actual real-time imbalances.Source: §6 · pp. 25–26
  • A fully coupled model copes with power-supply uncertainty and uses natural-gas-system flexibility, while linepack comparisons highlight the value of representing that flexibility.The authors also find that adding a flexibility source such as linepack is not adequate without market design that can operate it.Source: §6 · p. 26
  • Future work includes systematically defining coordination parameters and modeling compressor fuel consumption, which would increase computational complexity.The analysis indicates that systematic parameter definition can reduce expected system cost; tailored decomposition techniques may address the added computational burden.Source: §6 · p. 26

Parameters

The parameter glossary defines costs, capacities, demands, network limits, and scenario quantities used in the market-clearing models. It covers both electricity and natural-gas system components.

Parameters

  • Scenario-related parameters include scenario probabilities, expected and scenario-specific stochastic-plant output, and constants for gas-flow and pipeline linepack modeling.The glossary also defines the power-to-gas conversion factor for gas-fired plants and the linearization pressure points.Source: Parameters · pp. 26–27
  • Offer and regulation prices, along with electricity and gas load-shedding costs, define the cost parameters for dispatch and balancing.Day-ahead offers are listed for dispatchable power plants, gas producers, and storage; regulation offers are listed for plants and gas producers.Source: Parameters · pp. 26–27
  • Electricity and natural-gas demand, network limits, storage bounds, and production capacities are specified by node, plant, line, or period as appropriate.The glossary includes transmission capacity, pressure bounds, gas-storage injection and withdrawal rates, and reserve limits.Source: Parameters · p. 27

Sets

The sets define the model’s scenario, optimization-variable, plant, network, storage, and time-period collections. They also identify subsets and locations used to index system components.

Sets

  • The model’s core collections include stochastic-power scenarios, primal optimization variables, time periods, and fixed pressure points for linearizing the Weymouth equation.The glossary also enumerates the electricity and gas network branches and nodes.Source: Sets · pp. 27–28
  • Location-indexed sets group gas-fired plants, gas producers, storages, and electricity plants by their respective network nodes.Stochastic and dispatchable plants are indexed at electricity nodes; gas-fired plants, producers, and storages are indexed at gas nodes.Source: Sets · p. 27
  • Plant and storage sets distinguish dispatchable and stochastic power plants, gas-fired plants, gas producers, and natural-gas storages.The glossary also identifies thermal power plants as a subset of dispatchable plants.Source: Sets · pp. 27–28

Variables

The variable glossary distinguishes day-ahead quantities from scenario-specific balancing quantities across electricity and natural-gas systems. It also lists network states, flows, prices, and supply shortfalls.

Variables

  • Network and market variables include electricity and gas locational marginal prices, electricity-line power flows, gas-pipeline flows, and binary pipeline-flow directions.The glossary distinguishes day-ahead prices and flows from scenario-specific balancing prices and flows.Source: Variables · pp. 28–29
  • Scenario-indexed variables represent power-plant and gas-producer regulation, storage and pipeline flows, linepack, load shedding, and power spill.The glossary also includes scenario-specific electricity and gas prices and electricity voltage angles.Source: Variables · pp. 28–29
  • Day-ahead variables include plant dispatch, gas-producer output, storage and pipeline flows, pipeline linepack, nodal pressure, and storage volume.These quantities are indexed by period where specified.Source: Variables · pp. 28–29
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