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Linear Models for Optimization of
Interconnected Gas and Electricity Networks
Bjorn H. Bakken, Member IEEE, and Sigrun Kavli Mindeberg
Abstract— This presentation demonstrates a new natural gas
network module implemented in the optimization model
"eTransport", including nodes, pipes, valves and compressors.
A special linearization technique is used to keep track of the gas
pressure in a generic network structure. Together with models
for gas source and storage, gas-fired power plants and the
existing electricity network module, the new gas module makes
it possible to analyze interconnected natural gas and electricity
networks where both operational and structural (investment)
changes in one part of the system will influence capacities and
performance in the rest of the system.
Index terms-- Gas network, Electricity network, Linear
programming, Power system planning, Energy markets
I. INTRODUCTION - THE ETRANSPORT MODEL.
The optimization model "eTransport" is developed for
energy systems where several parallel energy carriers and
technologies are considered simultaneously [1, 2]. The
model uses a detailed network representation of
technologies and infrastructure to enable identification of
single components, cables and pipelines. eTransport
optimizes both operation and investments over a planning
horizon of 10 to 30 years for most relevant energy carriers
(electricity, gas, heat, biomass etc) and conversion between
these. It is not limited to continuous transport like lines,
cables and pipelines, but can also include discrete transport
by ship, road or rail.
The model is separated into an operational model (energy
system model) and an investment model [3]. In the
operational model there are sub-models for each energy
carrier and for different conversion components. The
operational planning horizon is relatively short (1-3 days)
with a typical time-step of one hour. The operational model
finds the cost-minimising diurnal operation for a given
infrastructure and for given energy loads. Annual operating
costs for different energy system designs are calculated by
solving the operational model repeatedly for different
seasons (e.g. peak load, low load, intermediate etc), different
investment periods (e.g. 5 year intervals) and relevant
system designs. Annual operating and environmental costs
for all different periods and energy system designs are then
used by the investment model to find the investment plan
that minimizes the present value of all costs over the
planning horizon.
__________________________
This work is sponsored in part by the Research Council of Norway.
Bjorn H. Bakken and Sigrun K. Mindeberg are with SINTEF Energy
Research, Trondheim, NO-7465, Norway
(e-mail: bjorn.h.bakken@sintef.no).
To enable a simultaneous optimization of several parallel
infrastructures, a generalized network structure is developed.
The sub-models for different components are connected by
general energy flow variables that identify the flow between
energy sources, network components for transport,
conversion and storage, and energy sinks like loads and
markets. The connections between these are case-specific,
and are identified by sets of pairs where each pair shows a
possible path for the energy flow between component types.
General energy flow variables are defined over the energy
system structure to account for the actual energy flow
between different components (except for internal flow
within each model). These general variables are included in
and restricted by the various models and they establish the
links between the different models.
In the operational model the different technology models
are added together to form a single linear optimization
problem where the object function is the sum of the object
functions from the different models, and the restrictions of
the problem include all the restrictions defined in the models.
Emissions are caused by a subset of components (power
plants/CHP's, boilers, road/ship transport etc) that are
defined as emitting CO2, NOx, CO and SOx. Further
environmental consequences can be defined. Emissions are
calculated for each module and accounted for as separate
results. However, if emission penalties are specified by the
user (e.g. a CO2 tax), the resulting costs are included in the
objective function and thus added to operating costs.
The task for the investment model is to find the optimal
set and order of investments during the period of analysis,
based on investment costs for different projects and the precalculated annual operating costs for different periods and
states. The optimal investment plan is defined as the plan
that minimizes the discounted present value of all costs in
the planning period, i.e. operating costs plus investment
costs minus the rest value of investments. The optimal plan
will therefore identify the optimal design of the energy
system (i.e. the optimal state) in different periods. More
details of the investment algorithm in eTransport can be
found in [3].
Mathematically, the model uses a combination of linear
programming (LP) and mixed integer programming (MIP)
for the operational model, and dynamic programming (DP)
for the investment model. The operational model is
implemented in the AMPL programming language with
CPLEX as solver [4], while the investment model is
implemented in C++. A modular design ensures that new
technology modules developed in AMPL for the operational
model are automatically embedded in the investment model.
A full-graphical Windows interface is developed for the
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model in MS Visio. All data for a given case are stored in a
database and can be accessed by the user through the GUI.
A special sub-module for CO2 capture and storage (CCS)
technologies has recently been published [5]. The models
for the CCS infrastructure are formulated consistent with the
models for gas, electricity and heat infrastructures. This has
introduced the flow of mass [tonne/h] in parallel with the
flow of energy [MWh/h]. Furthermore, the CCS models are
able to draw energy from the rest of the network to operate
compressors, pumps, absorbers etc.
Currently, work is performed in the operational model on
advanced biomass modules with natural long-term drying
and decaying processes, and in the investment model on
stochastic optimisation with uncertain prices (both fuel and
technologies) and uncertain energy demand.
III. SUMMARY
In this presentation a new natural gas network module
implemented in the optimization model "eTransport" is
demonstrated. The new module includes sources, nodes,
pipes, valves, compressors, loads and markets. A special
linearization technique is used to keep track of the gas
pressure in a generic network structure. The new gas module
is formulated consistent with existing electricity and heat
network modules, so the model can be used to analyze both
operation and investments in a complex interconnected
energy system with multiple suppliers, markets and loads.
IV. REFERENCES
1.
II. GAS NETWORK MODULE
The original gas pipeline model implemented in
eTransport was a point to point pipeline for bulk
transmission of large amounts of natural gas. This model did
not allow representation of more complex gas
infrastructures, however, so it has been replaced by a
completely new module with pipelines, compressors and
valves interconnected through a flexible nodal structure. In
the new network structure each gas component must be
connected to a node that keeps track of the pressures in the
system. The pressure in all components connected to one
specific node has to equal the pressure in this node. This
structure makes it possible to model a general natural gas
network where several sources, pipelines, compressors,
valves and markets/customers are interconnected.
The nodal structure of the gas infrastructure is based on
piecewise linearization of the Weymouth equation [6, 7, 8].
One of the advantages of the Weymouth equation is that the
friction coefficient (transmission factor) is only given by the
inner diameter of the pipe. In [6] the friction factor used in
the Weymouth equation is compared with among others the
friction factor used in the AGA Fully Turbulent Equation. It
is shown that the Weymouth equation overestimates the
friction factor when compared to AGA, but at higher
diameters (15” and above) the difference is small. The
linearization is done around a set of L pairs of inlet and
outlet pressures (PIi, POi) for each compressor and pipeline.
The user specifies the number of linearization pressure
points between the maximum/minimum values of inlet and
outlet pressure defined for the component, and the pressure
pairs are then generated automatically in the model.
2.
3.
4.
5.
6.
7.
8.
Bakken BH, Wolfgang O, Roynstrand J, Frydenlund F, Skjelbred HI:
"eTransport: A novel tool for energy system planning", Technical
Report A6255, SINTEF Energy Research, Trondheim, Norway, 2006,
ISBN 82-594-2966-7
Bakken BH, Holen AT: "Energy service systems: Integrated planning
case studies", in Proc. of the IEEE PES 2004 General Meeting,
Denver, CO, USA, 2004.
Bakken BH, Wolfgang O, Skjelbred HI: "eTransport: Investment
planning in energy supply systems with multiple energy carriers",
Energy 32 (2007), pp 1676–1689
Fourer R, Gay DA, Kerninghan BW: AMPL: A Modeling Language
for Mathematical Programming, Toronto, Thomson, 2003
Bakken BH, Velken, I: "Linear Models for Optimization of
Infrastructure for CO2 Capture and Storage", in IEEE Transactions on
Energy Conversion 23 (2008), pp 824-833
P. M. Coelho and C. Pinho, "Considerations about equations for
steady state flow in natural gas pipelines," in Journal of the Brazilian
Society of Mechanical Sciences and Engineering, vol. 29 (2007), pp.
262-273
Tomasgard A, Bjørndal M, Midthun K: "Modeling optimal economic
dispatch and flow externalities in natural gas networks", in Proc. of
the Trans-Atlantic Infraday Conference on Applied Infrastructure
Modeling and Policy Analysis, Maryland, USA, 2007
Midthun, K: Optimization models for liberalized natural gas markets,
Doctoral thesis, Norwegian University of Science and Technology,
Trondheim, Norway, 2007, ISBN 978-82-471-4526-5
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