SoftwareX 16 (2021) 100870
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SoftwareX
journal homepage: www.elsevier.com/locate/softx
Original software publication
Power Market Tool (POMATO)
for the analysis of zonal electricity markets
Richard Weinhold a,∗,1, Robert Mieth b,c,1
aFakultät VII Wirtschaft und Management, TU Berlin, 10623 Berlin, Germany
bDepartment of Electrical and Computer Engineering, Tandon School of Engineering, New York University, New York, NY 10012, USA
cFakultät IV Elektrotechnik und Informatik, TU Berlin, 10587 Berlin, Germany
article info
Article history:
Received 23 November 2020
Received in revised form 17 June 2021
Accepted 19 October 2021
Keywords:
Flow-based market coupling
FBMC
Economic dispatch problem
Transmission system
Optimal power flow
Security constrained optimal power flow
abstract
The proposed open-source Power Market Tool (POMATO) aims to enable research on interconnected
modern and future electricity markets in the context of the physical transmission system and its
secure operation. POMATO has been designed to study capacity allocation and congestion management
policies of European zonal electricity markets, especially flow-based market coupling (FBMC). For this
purpose, POMATO implements methods for the analysis of simultaneous zonal market clearing, nodal
(N-k secure) power flow computation for capacity allocation, and multi-stage market clearing with
adaptive grid representation and redispatch. The computationally demanding N-k secure power flow is
enabled via an efficient constraint reduction algorithm. POMATO provides an integrated environment
for data read-in, pre- and post-processing and interactive result visualization. Comprehensive data
sets of European electricity systems compiled from Open Power System Data and Matpower Cases
are part of the distribution. POMATO is implemented in Python and Julia, leveraging Python’s easily
maintainable data processing and user interaction features and Julia’s well readable algebraic modeling
language, superior computational performance and interfaces to open-source and commercial solvers.
©2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Code metadata
Current code version 0.3
Permanent link to code/repository used for this code version https://github.com/ElsevierSoftwareX/SOFTX_2020_79
Code Ocean compute capsule
Legal Code License LGPL v3
Code versioning system used git
Software code languages, tools, and services used Python and Julia
Compilation requirements, operating environments & dependencies Python and Julia, OS independent
(continuous automated tests on Windows and Ubuntu)
If available Link to developer documentation/manual https://pomato.readthedocs.io/en/latest/
1. Motivation and significance
Europe’s increasing electricity production from renewable en-
ergy resources in combination with a significant decline of
conventional generation capacity, has spawned political and
∗Corresponding author.
1R. Weinhold and R. Mieth contributed equally to this work.
academic interest in the transmission system’s ability to accom-
modate this transition [1]. Specifically, higher levels of coordi-
nation in a pan-European electricity market to facilitate meet-
ing emission targets, while ensuring reliable power supply at
acceptable cost [2], require the internalization of the power
transmission capabilities into trading processes. Central to this
discussion is the efficiency of capacity allocation and conges-
tion management (CACM) policies between and within electricity
market areas that share transmission infrastructure [3]. Capacity
allocation (CA) summarizes regulatory and market mechanisms
that constrain electricity trading volumes between two adjacent
https://doi.org/10.1016/j.softx.2021.100870
2352-7110/©2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Richard Weinhold and Robert Mieth SoftwareX 16 (2021) 100870
market areas with respect to the expected available cross-border
transmission capacity at time of delivery. Congestion manage-
ment (CM), on the other hand, refers to methods ensuring that
the physical system state at time of delivery indeed remains
within its security margins, e.g., does not cause transmission line
overloads. Noticeably, well defined CA based on suitable forecasts
can reduce CM measures, such as out-of-market generator re-
dispatch coordinated by the transmission system operator (TSO).
Previously implemented CA policies based on static net-transfer
capacities (NTCs) consider tie-line capacities between markets,
but neglect restricting transmission assets within market zones,
thus leading to often non-feasible market outcomes, increased
CM (redispatch) or overly conservative market results [4,5]. To
overcome these deficiencies and in an effort to ‘‘move towards a
genuinely integrated [European] electricity market’’ [3], the central-
western European (CWE) countries inaugurated flow-based mar-
ket coupling (FBMC) [6], a more complex CA policy that aims to
increase the potential volume of cross-border electricity trading
while decreasing CM requirements by explicitly accounting for
cross-border and zone-internal transmission limits.
The proposed Power Market Tool (POMATO) has been designed
to enable further research on the status-quo and future policies
of practical zonal electricity markets, especially FBMC. While
the theory of a centrally coordinated zonal electricity market is
somewhat mature, see e.g. [7], practical implementation with
imperfect coordination between market and system operators
requires ongoing analyses. The current European FBMC is a multi-
stage process coordinated by multiple TSOs and involves detailed
zone-specific load and generation forecasts and network mod-
els, which are typically not, or only partially, disclosed by the
TSOs [8]. However, the medium- and long-term evolution of the
FBMC design requires public decision support from independent
FBMC analyses, that, e.g., study the impact of more countries
joining the coupled market or bidding zone layouts (which have
been declared inefficient by a recent TSO study [9]). Notably, the
yearly federal report on the future of the grid in Germany (‘‘Net-
zentwicklungsplan’’), included a rudimentary FBMC representation
for the first time in its 2018 edition [10], three years after FBMC
implementation.
There are few model implementations of the FBMC process
available to the academic community [11–13]. In [11] Aravena
et al. compare various CA policies and demonstrate the benefits
of FBMC traditional approaches using real-world data sets. Also,
this study proposes an iterative approach to meet the practical
requirement that any dispatch has to be robust against unplanned
transmission equipment outage (N-1 security). However, data-sets
and model implementation are not published along the paper.
Similarly, Matthes et al. [12] present a FBMC formulation for
estimating future FBMC parameters and studying the impact of
regulations that require additional security margins on critical
transmission lines. This model uses an external convex hull re-
duction for a N-1 secure FBMC solution and is implemented in
the MILES framework [14], which, to our knowledge, is also not
publicly available. Schönheit et al. [13] extend the classic GAMS2
model ELMOD [15] to facilitate FBMC analyses and study the im-
pact of regulations requiring a minimum availability of interzonal
trading volumes [16]. Here, N-1 security is approximated through
static security margins. Despite comprehensive documentation
in [13] the code extension itself is undisclosed.
POMATO aims to overcome some of the caveats of [11–13]
by providing a documented and open-source framework that is
easy to use and goes beyond the implementation of an academic
optimization model. Some main features and contributions are:
2www.gams.com.
(a) Separation of data processing (implemented in Python3)
and optimization (implemented in Julia4) to achieve a flex-
ible Python-based user interface that is familiar to many
users, while creating a lean implementation of the central
optimization model in the well-readable JuMP algebraic
modeling language [17].
(b) Open-source available and documented on GitHub5[18,
19].
(c) Comprehensive data set of the European electricity mar-
ket and transmission infrastructure based on Open Power
Systems Data6and Matpower7data sets.
(d) Electricity market model with zonal and nodal market
clearing and a module to synthesize the FBMC process,
including heat sector coupling.
(e) Exact N-1 secure dispatch implementation suitable for
large-scale networks and multi-period analyses. The used
algorithm removes redundant constraints, similar to the
convex hull approach in [12], but has been optimized
for optimal power flow (OPF) analyses, yielding improved
computation times.
(f) Stochastic OPF using chance-constraints to analyze the im-
pact of forecast errors from renewable energy sources.
2. Software description
POMATO’s architecture is structured in three layers as shown
in Fig. 1. The model core collects the mathematical formulations
of the necessary optimization problems and provides an interface
to the required solvers. To ensure a lean model implementation
and efficient re-runs without re-calculating large parameter sets,
the model core is encapsulated in a data processing layer. This
layer automates parameter calculation and validation, provides
parameters to the model core, and validates and processes the
resulting model output. Finally, all functionality of POMATO is
collected in the outer user interface layer via readable API-like
commands.
2.1. Model core
The core of POMATO is an electricity market model. This model
is a linear optimization problem (LP) that finds the least cost
generation schedule that matches system demand with respect
to all technical constraints. Eq. (1) shows the general structure of
such a model8:
min OBJ =∑COST_G+COST_H+COST _CURT +OOM_PEN
(1a)
s.t.
Cost definition (1b)
Generation constraints (1c)
Heat constraints (1d)
Storage constraints (1e)
Energy balances (1f)
Network constraints (1g)
3www.python.org.
4www.julialang.org.
5www.github.com/richard-weinhold/pomato.
6www.open-power-system-data.org/.
7www.matpower.org/.
8The exact mathematical equations can be found in the pomato documen-
tation under http://pomato.readthedocs.io/en/latest/model_formulation.html. See
also in [8,20].
2
Richard Weinhold and Robert Mieth SoftwareX 16 (2021) 100870
Fig. 1. POMATO’s layered architecture. The inner model core is colored purple, the data processing layer is colored yellow, and the outer user interface is colored
blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Objective (1a) minimizes the cost of electricity generation
(COST_G), heat generation (COST_H), load curtailment
(COST_CURT ) and out-of market penalties (OOM_PEN, e.g. cost
of redispatch) over all units and time steps. Constraints in (1b)
define these cost components as sets of linear functions. Depend-
ing on the modeled market framework, some cost may be zero.
Constraints (1c) and (1d) enforce limits on the available elec-
tricity and heat generation, respectively. Constraints (1e) enforce
the time-coupled energy constraints for storages and limit their
efficiency-corrected charging and discharging power. Eqs. (1f) en-
sure that for all market areas generation equals demand plus net
export (export minus import) at all times. Network constraints
(1g) capture the transmission system topology as nodes (buses)
and lines and enforce exchange capacity limits. Physical flows
are modeled using the linearized (DC) power flow equations so
that nodal net injections can be mapped to power flows either
directly using power transfer distribution factor (PTDF) matrix or
implicitly via auxiliary voltage angle variable, see e.g. [21,22].
The specific definitions of constraints (1b)–(1g) are activated
and parametrized by POMATO ‘‘on the fly’’ based on user-defined
options. For example, nodal market clearing enforces an energy
balance in (1f) for each node and exchanges are limited by the
physical power flow and the thermal transmission system ca-
pacities. Zonal markets, on the other hand, ensure an aggregated
energy balance for an entire zone (i.e. multiple connected nodes).
Exchanges with neighboring zones are limited either explicitly
by NTCs, which constrain individual cross-border exchanges, or
implicitly through constraints on the net positions (total net
export) of each zone. The latter is used to model FBMC by cal-
culating zonal PTDFs, see e.g. [8], that map zonal net positions to
power flows on (critical) lines. Additionally, constraints to model
CM (redispatch) and external security requirements (contingency
robustness) can be included. For redispatch models, a nodal mar-
ket is solved subject to additional out-of-market penalties based
on results of a previously solved zonal market. Specifically, the
optimal redispatch is the least-cost change from a given genera-
tion schedule with predefined out-of-market penalties and fixed
zonal balances so that the resulting nodal balances are feasible
for the given transmission network. Preventive system security
requirements based on contingency (N-1) analyses, are enforced
through a suitable extensions of zonal and nodal PTDF matrices.
Optimal dispatch with preventive robustness against all con-
tingencies (unplanned outages) is computationally demanding
and requires preliminary solution heuristics [21]. Otherwise, be-
cause the number of constraints grows exponentially with the
network size, typical multi-period dispatch problems for real-
world economic analyses cannot be solved in reasonable time
or with commonly available computation resources. However, it
has been shown that many (in fact most) of these constraints are
redundant, i.e. never binding in the optimal solution due to the
existence of more restrictive constraints [23]. To ensure feasible
solution times for real-world networks over longer time horizons
(e.g. weeks to years), POMATO’s model core includes additional
functionality to identify these redundant constraints. Implemen-
tation and performance of this redundancy-removal algorithm
has been presented in our previous work [21], where we showed
that for most of the studied transmission networks over 99% of
the constraints can be removed from the optimization problem.
This reduced the solve time of smaller problems significantly and
enabled the solution of previously infeasible larger models.
The methods and algorithms implemented in the model core
require a suitable framework that enables fast robust computa-
tions and facilitates the use of interchangeable solver back-ends.
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Richard Weinhold and Robert Mieth SoftwareX 16 (2021) 100870
The open-source Julia Language provides a competitive combi-
nation of performance and readability [24], as well as suitable
libraries for implementing the required mathematical program-
ming (electricity market model and redundancy removal). Here,
we opted for the popular, well-readable and flexible JuMP pack-
age [17]. POMATO’s Julia modules (MarketModel.jl and Redun-
dancyRemoval.jl) are parametrized and called automatically
by the higher POMATO layers, as indicated in Fig. 1.
Feature Remark 1 (Risk-Aware Economic Dispatch).When model-
ing systems with high shares of stochastic renewable generation,
feed-in uncertainty with respect to capacity reserve, network
constraints and risk-perception of the system operator can be
internalized in the marker clearing using chance constraints [25,
26]. This capability is released with POMATO as an experimental
feature. See [20] for additional information.
Feature Remark 2 (Solvers).JuMP allows to easily communicate
with a wide range of solvers. Per default, POMATO will install
and use the open CLP solver.9However, if available on the user’s
system, POMATO can be configured to use commercial solvers
like Mosek10 or Gurobi.11 Other solvers can be implemented as
outlined in the documentation.12
2.2. Data management and pre-processing
POMATO’s central Data-Management layer, see Fig. 1, manages
the functionality of the model core by providing data and model
parametrizations, calling and monitoring the Julia processes, and
validating their results. To enable flexible data-handling that
avoids redundant calculations, we leverage Python and its object-
oriented programming paradigm, which resolves issues typically
related to static scripts. In such linear approaches, e.g. in GAMS,
all parameters are handled in the global scope of the program so
that any desired conditional functionality renders the code overly
complex and hard to maintain. Alternatively, encapsulating the
required functionally in different objects (modules), ensures data
consistency and compatible methods.
The data-management layer consists of four main modules,
DataManagement,GridTopology,GridModel and Market-
Model. These modules are created once in every instance of
POMATO and each provides specific methods to the user-interface
and other modules. Additionally, single-use modules are used for
result processing, e.g. FBMC calculations and result visualization.
DataManagement is the central module to handle all data and
their corresponding options. All input data is validated with
respect to predefined data structures that are required to run the
desired market model and are made available to other modules.
Due to the persistence of DataManagement, changes in data,
e.g. from user manipulations or model results, are propagated
throughout POMATO.
9www.github.com/jump-dev/Clp.jl.
10 www.github.com/MOSEK/Mosek.jl.
11 www.github.com/jump-dev/Gurobi.jl.
12 http://pomato.readthedocs.io/en/lastest/installation.html#solvers.
Pre-processing and validation of data related to the trans-
mission network topology requires specific methods that are
collected in the GridTopology module. It uses network data
(bus/node and line/branch information) and provides the pa-
rameters for subsequent power flow calculations. Additionally,
GridTopology verifies the properties of the network graph,
sets reference nodes for multiple disconnected networks, creates
contingency scenarios and manages settings of phase shifting
transformers.
The verified and pre-processed data is then used by the Grid-
Model and MarketModel modules to provide the input data for
the model core. Based on the market model that is desired from
the user (and that is feasible given the available data), GridModel
uses the data provided from GridTopology and DataManage-
ment to compile the correct grid representation with all physical
properties. This includes the parametrization of the network con-
straints, as discussed in Section 2.1 above, as well as generator
locations, cost models, and zonal configurations. Also, GridModel
calls RedundancyRemoval in the process of compiling the set
of parameters for a N-1 secure dispatch. If this algorithm has
been performed in previous runs, GridModel loads available
parameters to avoid time-consuming re-runs. These parameters
define the constraints under which the market has to be cleared.
Finally, MarketModel collects these constraints and manages the
required model runs. After the optimization in the model core
has obtained an optimal solution, MarketModel instantiates a
new Results object in the DataManagement module. Commu-
nication between the POMATO instance and the mathematical
core is achieved via a threaded process within the JuliaDaemon
module, see Remark 4.
The decoupling of model runs and data processing allows to
efficiently solve multi-stage market clearing processes that iterate
between different market model configurations. Two relevant
examples are itemized below.
Ex. 1 - Redispatch Analysis: Zonal electricity markets are cleared
as a single price zone without any internal network con-
straints. In a second stage, potential transmission line over-
loads are rectified by out-of-market CM measures, i.e., re-
dispatch. For this purpose, a second model run with nodal
resolution is required.
Ex. 2 - FBMC: FBMC uses a forecasted nodal dispatch (base case)
to derive zonal PTDF matrices that establish the relation
between zonal net-positions and flows on critical network el-
ements (CNEs). The resulting feasible region of net-positions
with respect to the technical limits of the CNEs are called
FBMC domains. These domains are used to solve the zonal
market clearing that is then corrected by subsequent CM
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Richard Weinhold and Robert Mieth SoftwareX 16 (2021) 100870
Fig. 2. Exemplary flow based domain.
measures. See also [20] for more information the FBMC pro-
cess. The supplementary FBMCModule automatizes the nec-
essary re-parametrization of parts of the model data, includ-
ing the generation of the FBMC domains depicted in Fig. 2, and
facilitates the necessary re-runs of the optimization model.
Feature Remark 3 (Contingency Scenarios).The user can specify
customized contingency scenarios instead of performing a strict
N-1 contingency analysis (i.e., protecting the system against the
outage of every single transmission line). Such scenarios can
either ignore certain lines as potential contingencies or define
contingency groups of outages that are likely to occur simulta-
neously, e.g., parallel lines.
Feature Remark 4 (Python–Julia Interface).Communication be-
tween the Julia language and Python is implemented by a
JuliaDaemon class that maintains a threaded Julia program
and executes its functions when requested by MarketModel or
GridModel. Instructions are transferred via json files that are
written to the POMATO package folder. This way, the Julia process
has to be started only once and remains alive after a model is
completed, which allows fast re-solving of a model with different
parameters.
2.3. User interface
All modules of the data processing layer presented in previous
Section 2.2 are attributed to the overall POMATO object, which
provides a comprehensive set of functions to the user. Although
the methods of the lower level modules are accessible to the user,
the POMATO module provides predefined methods for the typical
use cases of this program. A brief overview is presented in the
illustrative application in Section 3. Additionally, POMATO pro-
vides a collection of visualization features. Its create_geo_plot
method shows market and redispatch results including prices on
an interactive map using the Ploty13 library. Many results, such as
the FBMC domains or generation schedules, can be automatically
plotted as a collection of interactive 2D diagrams. In addition,
POMATO can start a Dash14 server, featuring a curated view of
interactive plots and visualizations. See documentation15
2.4. Publication
POMATO is available under the LGPLv3 license on GitHub in-
cluding a comprehensive documentation [18,19]. To facilitate
testing, maintenance and readability, pomato’s Python and Julia
components are stored in separate repositories. However, all
installation is handled by the central Python module in [18]. See
also Remark 5. The repository also includes exemplary data-sets
that cover the core functionality. These include a data-set for
the German power system, that has been compiled to showcase
POMATO’s functionality, as well as the NREL 118 bus network [27],
and Matpower case files. More details on how to create custom
data-sets can be found in the accompanying documentation.16
Feature Remark 5 (Installation).POMATO requires Python and
Julia to be installed. When both are available on the user’s system
the installation of POMATO is managed through Python via pip.17
The Python installer will detect the Julia distribution on the sys-
tem and automatically pull the two Julia packages, MarketModel
and RedundancyRemoval from their separate repositories. They
will be installed in a Julia virtual environment to isolate any
dependencies from possible user-installed Julia packages.
3. Illustrative application
This section illustrates the application of POMATO. The corre-
sponding example code is shown below and its line numbers are
referenced in the subsequent description.
13 www.plotly.com.
14 www.plotly.com/dash/.
15 http://pomato.readthedocs.io/en/latest/model_functionality.html#the-
dashboard.
16 http://pomato.readthedocs.io/en/latest/input_model_data.html.
17 www.pip.pypa.io.
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Richard Weinhold and Robert Mieth SoftwareX 16 (2021) 100870
1from pomato import POMATO
2
3mato = POMATO(
4wdir=wdir,
5options_file=" options/de.json "
6)
7mato.load_data(’data_input/DE_2020.zip’)
8
9mato.create_grid_representation()
10 mato.run_market_model()
11
12 # Return market results (m)
13 # and redispatch results (r)
14 m, r = mato.data.return_results()
15
16 # Check for Overloaded lines N-0
17 n0_m, _ = m.overloaded_lines_n_0()
18 print(" Number of N-0 Overloads: " ,
19 len(n0_m))
20 n0_r, _ = r.overloaded_lines_n_0()
21 print(" Number of N-0 Overloads: " ,
22 len(n0_r))
23 n1_r, _ = r.overloaded_lines_n_1()
24 print(" Number of N-1 Overloads: " ,
25 len(n1_r))
26
27 gen = r.redispatch()
28 print(" Total Redispatch in MWh: " ,
29 gen.delta_abs. sum())
30
31 mato.visualization.create_geo_plot(
32 redispatch_result,
33 show_prices=True ,
34 show_redispatch=True)
After importing the POMATO module in line 1, a new POMATO
class is instantiated in line 3with the requested working di-
rectory (wdir) and an options file in the JSON18 format that
configures the model run. In this example the options file spec-
ifies a zonal market clearing, subject to NTCs, and subsequent
redispatch with redispatch-cost and model horizon:
{"optimization": {
" type " :" ntc " ,
" model_horizon " : [0, 168],
"redispatch": {
" include " : true,
" cost " : 50}}
A comprehensive list and descriptions of possible configu-
rations can be found in the online documentation.19 In line 4
the POMATO object loads the required data, which automati-
cally instantiates the data management objects as described in
Section 2.2 above. The command in line 6creates the grid rep-
resentation for the specified model run and line 7starts the
actual model run. After successful completion, two Results are
instantiated into DataManagement and are available to the user
(see line 9) for further analysis, e.g., to check for overloaded
lines with and without contingencies (see lines 11–20). Finally
a geo-plot is created in line 22, which is shown in Fig. 3.
18 www.json.org.
19 http://pomato.readthedocs.io/en/latest/options.html.
Execution time of this example is around 7.5 min, including
data read-in, Julia start-up, solving both the market- and the
redispatch problem, as well as the final result analysis and plot-
ting. Subsequent runs benefit from already active Julia processes,
which saves 2 to 4 min of computation time.
4. Impact
POMATO offers features that may be of interest for practition-
ers, researchers and students alike, all of which benefit from
POMATO’s open-source availability.
•The high-level API-like functionality of the top-layer PO-
MATO class (see Section 2.3) facilitates the use of POMATO
without a detailed understanding of power engineering and
economics. With minimal understanding of the required
data structure, POMATO offers a low barrier for users to
obtain relevant results from their own data sets and to
produce insightful visualizations of these results.
•By separating the model core from the user interface, more
advanced users will be able to extend or modify the math-
ematical formulations for their needs, which we hope to
enable new research in the area of zonal market clearing.
•Advanced interactive visualizations can support educators
to convey some concepts of power system operations and
electricity markets.
5. Conclusion and further development
This paper presents a tool that aims to enable the broad
analyses of zonal electricity markets. The proposed Power Mar-
ket Tool POMATO is open-source, provides essential data-sets
and comprehensive online documentation. Central to POMATO’s
functionality is its ability to synthesize the flow-based market-
coupling (FBMC) process. Further, POMATO includes a redundancy
removal algorithm that enables security constrained (N-k) dis-
patch with feasible computation times. While the current version
of POMATO already includes prototypical options for the analyses
of stochastic (chance-constrained) market clearing, extensions to
stochastic and risk-aware markets and power flow analyses along
the lines of [25,26] are subject to ongoing development.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared
to influence the work reported in this paper.
Acknowledgements
This work was supported by the German Federal Ministry
for Economic Affairs and Energy (BMWi) in the project LKD-EU
(03ET4028A) and European Union’s Horizon 2020 research pro-
gram (No. 773406). The work of Robert Mieth was supported by
the Reiner Lemoine-Foundation. The work of Richard Weinhold
was supported by the Danish Energy Agency.
6
Richard Weinhold and Robert Mieth SoftwareX 16 (2021) 100870
Fig. 3. Geo-plot of market- and redispatch result for Germany. Red and green circles indicate negative and positive changes in generation schedules, respectively,
due to redispatch. Change magnitude are proportional to the circles’ radius. Line colors indicate average line load where green is low loads and red is high load.
Prices as dual multipliers on the nodal power balances of the redispatch run are indicated by the contour plot overlay. (For interpretation of the references to color
in this figure legend, the reader is referred to the web version of this article.)
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