# Design of JiuTian Intelligent Network Simulation Platform

1st Lei Zhao  China Mobile Research Institute
Beijing, China
<EMAIL_ADDRESS>2nd Miaomiao Zhang  China Mobile Research Institute
Beijing, China
<EMAIL_ADDRESS>3rd Guangyu Li  China Mobile Research Institute
Beijing, China
<EMAIL_ADDRESS>4th Zhuowen Guan  China Mobile Research Institute
Beijing, China
<EMAIL_ADDRESS>5th Sijia Liu  China Mobile Research Institute
Beijing, China
<EMAIL_ADDRESS>6th Zhaobin Xiao  China Mobile Research Institute
Beijing, China
<EMAIL_ADDRESS>7th Yuting Cao  China Mobile Research Institute
Beijing, China
<EMAIL_ADDRESS>8th Zhe Lv  China Mobile Research Institute
Beijing, China
<EMAIL_ADDRESS>9th Yanping Liang  China Mobile Research Institute
Beijing, China
<EMAIL_ADDRESS>
###### Abstract

This paper introduced the JiuTian Intelligent Network Simulation
Platform111JiuTian Intelligent Network Simulation Platform, which can provide
wireless communication simulation data services for the Open Innovation
Platform. The platform contains a series of scalable simulator
functionalities, offering open services that enable users to use reinforcement
learning algorithms for model training and inference based on simulation
environments and data. Additionally, it allows users to address optimization
tasks in different scenarios by uploading and updating parameter
configurations. The platform and its open services were primarily introduced
from the perspectives of background, overall architecture, simulator, business
scenarios, and future directions.

###### Index Terms:

Wireless communication, simulation platform, emulators

##  I Introduction

The application of artificial intelligence technology to solve technical and
application problems in the communication field and vertical industries has
become a mainstream direction[1, 2, 3]. However, the communication network
system is robust, complex, and has a long industrial chain. The intersection
of communication and AI poses significant challenges. Additionally, the
communication industry lacks a flexible and adaptable real-world verification
environment, which hinders the iterative testing for the development and
validation of network AI capabilities.

To address these challenges, the JiuTian Intelligent Network Simulation
Platform(JINSP) provides an implementation solution for innovative smart
networks, offering services in four dimensions: dynamic network simulation,
model training, capability invocation, and capability evaluation. By creating
virtual objects in the digital world to represent communication network
entities, people, network services, and their topological relationships, it
can effectively simulate, respond to, and predict the states and behaviors of
various entities in the physical environment. This facilitates the incubation
and breakthrough of network intelligence technology.

The JINSP provides various types of simulation services for research and
application personnel in the ”AI + communication” field and releases
corresponding research tasks. The platform aims to simulate and construct a
simulation environment for wireless communication networks, industry-specific
networks, etc., of a certain scale. This environment consists of different
business scenarios and can serve scenarios such as intelligent network
operation, maintenance, optimization, and service provision, as well as the
development, testing, and validation of AI capabilities for network element
intelligence.

##  II Related work

The JINSP follows the design principles and implementation logic of some
commonly used AI platforms and simulation software in the industry. It is
dedicated to providing an open-service platform for the wireless field. The
following is a comparative analysis between the intelligent network simulation
platform and several types of platforms currently existing in the industry.

  * •

AI General Cloud Services: This type of platform’s service capabilities are
mostly built on its own core business and provide a wide range of interface-
based applications for the general audience. Users can obtain inference
results by calling the API of the AI cloud service platform. Some well-known
examples include Google Cloud AI Platform, OpenAI API, Baidu AI Open Platform,
Tencent Cloud AI Lab, and so on. The original intention of the intelligent
network simulation platform design is also based on existing core businesses,
aiming to achieve platformization of services, and support user interaction
and capability openness.

  * •

Online Training Platforms: This type of platform supports mainstream algorithm
frameworks and provides GPU computing resources as well as some task datasets.
Its capabilities cover academic areas such as deep learning, computer vision,
natural language processing, etc.

This category of platforms is favored by educators and researchers due to its
superior hardware environment and user-friendly interface. However, these
platforms primarily serve to provide GPU computing services, and only a small
portion of datasets applicable to communication directions. They have a
limited impact on research in the ”AI + communication” field and still cannot
address the issues of limited data and lack of interactive network
environments in communication research.

  * •

Toolkits: This type of platform does not rely on interfaces for online
inference, nor does it provide a trainable platform. Instead, it offers the
complete core product and functionality to the user. Users can download or
reference the toolkit to create new instances locally, with each instance
tailored to meet individual user needs. Users can then use this as a basis for
training or inference. For example, with OpenAI’s Gym[4] platform, developers
can easily build and train intelligent agents [5, 6, 7] and evaluate their
performance.

  * •

Simulation Platforms: This category of platforms focuses on the simulation
implementation in specific vertical domains and does not inherently include AI
capabilities.[8] The simulation effects are achieved through underlying
engines. Some examples include Atoll, Planet, Exata, MATLAB, etc. Atoll and
Planet specialize in specific types of network simulation, such as wireless
network planning and satellite communication system design. Exata supports
various network types and technologies, including wireless networks and sensor
networks. These platforms are suitable for professional users in related
fields. However, due to the lack of support for AI algorithms, their
scalability and user flexibility are relatively low.

MATLAB provides a rich set of mathematical and simulation toolboxes for
various simulation modeling and performance evaluation tasks. It also offers
the capability to incorporate extended algorithm functionalities. However,
using MATLAB as a platform can be more challenging as it requires a
corresponding programming background.

Compared to the various open platforms described above, the intelligent
network simulation platform designed in this paper integrates features such as
online training and inference capabilities, dynamic simulation functionality,
and future open API services. Additionally, it is capable of providing
specific AI capabilities for the wireless communication vertical domain’s
particular business needs.

##  III Framework

The JINSP constructs different types of simulators and integrates them based
on the requirements of business scenarios. This enables users to conduct
online model training and inference for corresponding open tasks. Users
immerse themselves in the simulation platform environment and interact with it
to invoke simulation capabilities and manage tasks. For instance, users can
issue simulation configurations to invoke specific simulation capabilities and
perform simulation functions. Subsequently, users can update simulation
configurations based on the simulation results returned by the environment
(e.g., antenna parameters, base station positions, etc.). They can iterate the
invocation of simulator capabilities to accomplish online model training and
inference. This process is similar to the concept of an intelligent agent [9,
10], which can adapt its understanding based on the environmental
configuration. This chapter will provide a detailed introduction to the basic
simulators and the process of combining them in Figure 1.

Figure 1: The Overall Architecture Diagram of the Intelligent Network
Simulation Platform, includes basic emulators, combination emulators and
business scenario. Besides, open service and AI block can provide the basic
service for the JINSP.

###  III-A Basic Emulator

  * •

User Behavior Simulation involves trajectory generation and business
generation. Trajectory generation utilizes generative algorithms such as
Variational Autoencoders (VAE) [11, 12, 13] and Generative Adversarial
Networks (GAN) [14, 15] to simulate user trajectories, achieving the
generation of latitude and longitude sequences at a daily level. Business
generation addresses the synthesis of user-level business traffic. It is based
on a Knowledge-Enhanced GAN model. This model takes user behavior
representation vectors and packet distribution characteristic vectors
extracted from user traffic data packets as inputs for knowledge enhancement.
Subsequently, it employs GAN to generate packet sequences.

  * •

Large-Scale Channel Simulation is based on the antenna pattern of any sub-
beam, combined with the location, antenna height, transmit power, mechanical
azimuth, tilt angle, geographical terrain, etc., simulate the large-scale
channel simulation results for each grid or user-specific cell with any type
of beam. Subsequently, this supports the calculation of relevant metrics for
base stations and terminals, such as Service Cell RSRP/SINR [16, 17] values,
neighboring cell RSRP/SINR values, and the calculation of the corresponding
beam IDs for the service cell and neighboring cells.

  * •

Channel Simulation is based on real geographical environments and considering
the regional characteristics of the simulation scenario (such as Indoor, Umi,
Uma, etc.), wireless channel modeling is conducted across regions and multiple
scenarios. Channel simulation achieves channel modeling by invoking the
foundational simulator, which includes Large-Scale Channel Simulation, and
combines it with small-scale fading (fast fading) simulation. In this process,
small-scale fading simulation employs a common statistical modeling approach,
thoroughly simulating the multipath effects generated by processes such as
direct propagation, reflection, diffraction, transmission, and diffuse
scattering in the transmission of electromagnetic waves. Finally, the results
of the large-scale fading model are combined with those of the small-scale
fading model to generate the simulated channel matrix.

  * •

Base Station/Terminal Simulation provides wireless protocol stack processes
and functional simulation for Radio Resource Control (RRC), Medium Access
Control (MAC), and Physical Layer (PHY) in system-level simulation [18].

In the RRC layer simulation, key processes like cell access,
handover/reselection, and wireless resource management are included. The MAC
layer simulation, conducted separately on both the base station and terminal
sides, models critical processes such as wireless resource scheduling, MIMO
[19], link adaptation, wireless resource mapping, uplink power control, and
Hybrid Automatic Repeat reQuest (HARQ). The PHY layer simulation, guided by
the functional processes in the above two layers, ultimately calculates
system-level network performance metrics like the number of connected users,
cell load, uplink/downlink transmission rates, and more.

Additionally, the Base Station/Terminal Simulation module can be combined with
map information, cell parameters, and beam configurations. With the aid of AI-
calibrated path loss calculation formulas, it computes the Service Cell
RSRP/SINR values under any sub-beam antenna pattern in wireless coverage
simulation scenarios.

###  III-B Combination Emulator

In the previous section, individual basic simulators are responsible for
simulating the fundamental capabilities within the network system, but they
may not meet the simulation requirements of complex business scenarios [20].
The platform combines multiple basic simulators and provides unified services
to meet the task opening and online model training requirements of different
business scenarios. Currently, the intelligent network simulation platform
provides three types of combination simulators:

  * •

Real Environment Dynamic User Protocol Stack Simulation: This combination
simulator is composed of user behavior simulation, base station simulation,
and terminal simulation. It provides users with metrics such as RSRP, SINR,
traffic, and rates at both user and cell granularity in a real environment.

  * •

Real Environment Dynamic User Coverage Simulation: To improve the efficiency
of base station and terminal simulation, this simulator primarily invokes the
physical layer simulation from the above simulators. It combines with user
simulation to output coverage metrics such as RSRP and SINR at both user and
cell levels.

  * •

Link-Level Channel Simulation: This combination simulator utilizes the core
capability provided by channel simulation. It combines with base station and
terminal physical layer protocol stack simulation to provide users with
frequency-domain channel response information at the Resource Element (RE)
level.

##  IV Business Scenario

###  IV-A Multi-objective antenna opt.

Multi-objective optimization aims to assist network operators in using the
simulation platform’s twin-modeling capability of real network situations.
This allows for a cost-effective understanding of the evaluation results of
various network metrics under different cell SSB antenna weight parameters
[21]. Consequently, it enables more convenient and efficient optimization
results in different regions and time periods, considering multiple
optimization objectives.

A wireless cellular network consists of multiple base stations, each with
several cells. These cells achieve coverage through the transmission of
Synchronization Signal Blocks (SSBs). In 5G, cells can adjust the azimuth,
tilt angle, horizontal beamwidth, and vertical beamwidth of eight sub-beams to
control the coverage range and radius of the cell. Users scan through each SSB
beam and select the one with the strongest coverage signal. The cell simulates
access metrics and rates based on factors such as the user’s coverage signal
and interference.

For the open task of multi-objective antenna optimization, considering user
mobility, the task involves adjusting the sub-beams of the cell, including
azimuth, tilt angle, horizontal beamwidth, and vertical beamwidth. The
objective is to jointly optimize the metrics [22, 23] of all cells within the
region, allowing for the optimization of SS-RSRP, SS-SINR, total number of
users, total traffic volume, and rates within a five-minute granularity. This
optimization task aims to achieve the overall best performance in the region,
encompassing $n$ cells and $k$ users. Additionally, by controlling the
weighting coefficients, the optimization task can be tailored to focus on
specific metrics, transforming the problem into a coverage optimization
problem. The output of this task includes the azimuth, tilt angle, horizontal
beamwidth, and vertical beamwidth of each cell within the region every five
minutes, ensuring that various metrics (as mentioned above) reach their
optimal values across the entire area within a specified time frame.

###  IV-B CSI

In large-scale MIMO systems, base stations (BS) are typically equipped with as
many as several hundred active antennas, serving multiple user equipments
(UEs). Accurate Channel State Information (CSI) [24] is crucial for harnessing
the performance gains of large-scale MIMO. The CSI for the downlink channel
needs to be obtained by UEs through downlink pilot estimation and then
transmitted back to the base station through a feedback link. The base station
utilizes the received feedback information for tasks like precoding and other
adaptive transmission optimizations. Deep learning-based methods can address
the CSI compression feedback issue in large-scale MIMO systems, efficiently
and accurately enhancing the feedback precision. There are also many promising
research directions in this area worthy of further investigation.

This task aims to harness the feature extraction and information compression
capabilities of AI. It involves training an AI model using channel
characteristic information. This model is then used to compress the channel
characteristic information at the UE side. The compressed information is
transmitted through the channel and recovered at the receiving end. The base
station uses the recovered information for tasks like precoding and other
adaptive transmission optimizations.

The ultimate goal of CSI compression feedback is to restore the channel state
information with as little loss as possible under a fixed compression bit
requirement. Smaller compression bits result in fewer resources being used for
transmission. However, this can lead to lower precision in model recovery,
resulting in lower feedback performance. This task requires designing a model
under a specific feedback bit vector, with the objective of achieving higher
restoration accuracy of the model with the smallest possible compression bits.

##  V Future decision

The platform starts from the interdisciplinary integration of communication
and artificial intelligence, selects key technologies and application issues
of concern to the academic and production fields, analyzes and extracts the
necessary simulation environment capabilities for algorithm technology
development corresponding to various problems. It has constructed an
intelligent network simulation service that integrates various functions such
as user behavior simulation, wireless network coverage simulation, base
station protocol stack simulation, and channel simulation, forming a
systematic network simulation environment. Through simulation performance
optimization, capability combination, and interface encapsulation, the
platform provides a good user experience for the external service opening of
simulated network environments and data. It also contributes important
infrastructure to the integration and development of communication and
artificial intelligence technologies.

To provide stronger support for technological breakthroughs and promote
industry development, the platform still needs to continuously accumulate
strength in the following directions: First, with the development of
generative artificial intelligence and large-scale model technologies,
artificial intelligence technology has gradually made significant
breakthroughs in many areas that were once considered unsuitable or
unfeasible. The future focus of the platform is how to use multimodal
communication network big data and large-scale model technology to build
network simulation capabilities. Second, digital twinning is an important
evolution direction of 6G communication technology. The platform will continue
to expand the business scenarios covered by simulation capabilities, achieve
end-to-end full-link simulation, and accelerate the construction and opening
of trial networks in existing networks. It will overcome the coexistence of
simulation environment and real network, providing a reliable basis for the
technical feasibility demonstration of digital twin networks. Third, to meet
the personalized research and production needs of users from all walks of
life, the platform will continuously improve the flexibility and freedom of
the network simulation environment. It will support more parameters
configuration such as user behavior, network equipment, and simulation models,
while actively promoting the decoupling of simulation capabilities and the
design of unified interfaces. It will mobilize industry partners to contribute
basic simulation capabilities and build a flexible and assembled network
simulation environment. It will promote the open source and open access of
high-quality resources in the industry, establish a collaborative and
innovative industry ecosystem, and provide strong support for the national
network power strategy.

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#  XGA: A module for the large-scale scientific exploitation of archival X-ray
astronomy data

David J<EMAIL_ADDRESS>Paul A. Giles  Kathy Romer
Violetta Korbina

††margin:  DOI: N/A Software •
https://github.com/DavidT3/XGA/https://doi.orgRepository •
https://xga.readthedocs.io/https://doi.orgDocumentation Submitted: N/A
Published: N/A License
Authors of papers retain copyright and release the work under a Creative
Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/https://doi.orgCC BY 4.0).

## Summary

The _XMM_ Cluster Survey (XCS, Romer et al., 2001) have developed a new Python
module, X-ray: Generate and Analyse (hereafter referred to as XGA) to provide
interactive and automated analyses of X-ray emitting sources observed by the
_XMM_ -Newton space telescope. XGA only requires that a set of cleaned,
processed, event lists has been created, and (optionally) that a source
detector has generated region lists for the observations. XGA is centered
around the concept of making all available data easily accessible and
analysable. The user provides information (e.g. RA, Dec, redshift) on the
source they wish to investigate, and XGA will locate all relevant observations
and generate all required data products. This approach means that the user can
quickly and easily complete common analyses without manually searching through
large amounts of archival data for relevant observations, thus being left free
to focus on extracting the maximum scientific gain. In the future, we will add
support for X-ray telescopes other than XMM (e.g. Chandra, eROSITA), as well
as the ability to perform multi-mission joint analyses. With the advent of new
X-ray observatories such as _eROSITA_ (Predehl et al., 2021), _XRISM_ (XRISM
Science Team, 2020), _ATHENA_ (Nandra et al., 2013), and _Lynx_ (Gaskin et
al., 2019), it is the perfect time for a new, open-source, software package
that is open for anyone to use and scrutinise.

## Statement of need

X-ray telescopes allow for the investigation of some of the most extreme
objects and processes in the Universe; this includes galaxy clusters, active
galactic nuclei (AGN), and X-ray emitting stars. This makes the analysis of
X-ray observations useful for a variety of fields in astrophysics and
cosmology. Galaxy clusters, for instance, are useful as astrophysical
laboratories, and provide insight into how the Universe has evolved during its
lifetime.

Current generation X-ray telescopes have large archives of publicly available
observations; _XMM_ -Newton has been observing for over two decades, for
instance. This allows for analysis of large amounts of archival data, but also
introduces issues with respect to accessing and analysing all the relevant
data for a particular source. XGA solves this problem by automatically
identifying the relevant _XMM_ observations then generating whatever data
products the user requires; from images to sets of annular spectra. Once the
user has supplied cleaned event lists (and optionally region files) an
analysis region can be specified and spectra (along with any auxiliary files
that are required) can be created.

Software to generate X-ray data products is supplied by the telescope teams,
and most commands require significant setup and configuration. The complexity
only increases when analysing multiple observations of a single source, as is
often the case due to the large archive of data available. XGA provides the
user with an easy way to generate _XMM_ data products for large samples of
objects (which will scale across multiple cores), while taking into account
complex factors (such as removing interloper sources) that vary from source to
source.

Figure 1: Demonstration of the view methods of the RateMap and Spectrum
classes, when applied to the Abell 907 galaxy cluster. Data from the _XMM_
EPIC-PN instrument of 0404910601 is used. _Left_ : A count-rate map with a
mask that removes contaminant sources (using XCS region information) and
applies an $R_{500}$ aperture. _Right_ : A spectrum generated for the
$R_{500}$ region with contaminants removed, and fit with an absorbed plasma
emission model using XSPEC.

## Features

XGA is centered around source and sample classes. Different source classes,
which represent different types of X-ray emitting astrophysical objects, all
have different properties and methods. Some properties and methods are common
to all sources, but some store quantities or perform measurements that are
only relevant to a particular type of astronomical source.

XGA also contains product classes, which provide interfaces to X-ray data
products, with built-in methods for analysis, manipulation, and visualisation.
The RateMap (a count rate map of a particular observation) class for instance
includes view methods (left hand side of Figure 1), methods for coordinate
conversion, and for measuring the peak of the X-ray emission. We also provide
classes for interacting with, analysing, and viewing spectra (see right hand
side of Figure 1), both global and annular; as such we can use XGA to
investigate both average properties and, in the case of extended sources, how
these properties vary radially. Similar procedures for image based analysis
are also available, where images (and merged images from all available data
for a given source) can be easily generated en masse, then combined with masks
automatically generated from supplied region files to perform photometric
analyses.

We also include a set of profile classes, with built-in viewing methods, and a
fitting method based around the emcee ensemble MCMC sampler (Foreman-Mackey et
al., 2013). Profiles also support storing and interacting with fitted models;
including integration and differentiation methods, inverse abel transforms,
and predictions from the model. An example of the utility of these profiles is
the galaxy cluster hydrostatic mass measurement feature; this requires the
measurement of 3D gas density profiles, 3D temperature profiles, gas mass, and
total mass profiles.

To extract useful information from the generated spectra, we implemented a
method for fitting models, creating an interface with XSPEC (Arnaud, 1996), a
popular X-ray spectral fitting language. This interface includes the ability
to fit XSPEC models (e.g. plasma emission and blackbody) and simplifies
interaction with the underlying software and data by automatically performing
simultaneous fits with all available data.

Figure 2: A flowchart giving a brief overview of the XGA workflow.  Figure 3:
A demonstration of the application of XGA to an artificial observation of a
simulated Illustris-TNG cluster created by another XCS tool (XCSim; Turner et
al. in prep). The simulated cluster was injected into the 0302351301 XMM
observation. Cross-hair is the peak position of the simulated cluster in the
middle image, dashed circle is the simulation-defined $R_{500}$ of the
cluster. Red and green regions indicate XCS detected sources from the original
observation. Left-hand-side: Zoomed view of the original combined PN+MOS1+MOS2
image. Middle: Zoomed view of the combined PN+MOS1+MOS2 image after the
simulated cluster was added at $z=0.35$. Right-hand-side: An view of an XGA
AnnularSpectra object generated from the simulated cluster.

## Existing software packages

To the knowledge of the authors, no software package exists that provides
features completely equivalent to XGA, particularly in the open source domain.
That is not to say that there are no software tools similar to the module that
we have constructed; several research groups including XCS (Lloyd-Davies et
al., 2011), XXL (Giles et al., 2016), LoCuSS (Martino et al., 2014), and the
cluster group at UC Santa Cruz (Hollowood et al., 2019) have developed
pipelines to measure the luminosity and temperature of X-ray emitting galaxy
clusters, though these have not been made public. It is also important to note
that these pipelines are normally designed to measure a particular aspect of a
particular type of X-ray source (galaxy clusters in these cases), and as such
they lack the generality and flexibility of XGA. Our new software is also
designed to be used interactively, as well as a basis for building pipelines
such as these.

Some specific analyses built into XGA have comparable open source software
packages available; for instance pyproffit (Eckert et al., 2020) is a recently
released Python module that was designed for the measurement of gas density
from X-ray surface brightness profiles of galaxy clusters. We do not believe
that any existing X-ray analysis module has an equivalent to the source and
sample based structure which XGA is built around, or to the product classes
that have been written to interact with X-ray data products.

The XSPEC (Arnaud, 1996) interface we have developed for XGA is far less
comprehensive than the full Python wrapping implemented in the PyXspec module,
but scales with multiple cores for the analysis of multiple sources
simultaneously much more easily.

## Research projects using XGA

XGA is stable and appropriate for scientific use, and as such it has been used
in several recent pieces of work; this has included an XMM analysis of the
eFEDS cluster candidate catalogue (Turner et al., 2021), where we produced the
first temperature calibration between XMM and eROSITA, a multi-wavelength
analysis of an ACT selected galaxy cluster (Pillay et al., 2021), and XMM
follow-up of Dark Energy Survey (DES) variability selected low-mass AGN
candidates (Burke et al., 2021).

There are also several projects that use XGA nearing publication. The first of
these is a hydrostatic and gas mass analysis of the redMaPPeR (Rykoff et al.,
2014) SDSS selected XCS galaxy cluster sample (Giles et al., 2022) and well as
the ACTDR5 (Hilton et al., 2021) Sunyaev-Zel’dovich (SZ) selected XCS sample
of galaxy clusters. This work also compares commonly measured X-ray properties
of clusters (the X-ray luminosity $L_{\rm{X}}$, and the temperature
$T_{\rm{X}}$ both to results from the existing XCS pipeline and from
literature, confirming that XGA measurements are consistent with previous
work. This process is repeated with XGA’s galaxy cluster gas and hydrostatic
mass measurements, again showing they are consistent with previous work. XGA’s
ability to stack and combine X-ray surface brightness profiles is currently
being used, in combination with weak lensing information from DES, to look for
signs of modified gravity in galaxy clusters. Finally, we intend to use XGA to
analyse artificial XMM observations of simulated galaxy clusters (see Figure
3) produced by another XCS tool (XCSim; Turner et al. in prep). This will
allow us to perform measurements of simulated cluster properties for various
simulation suites (e.g. Illustris-TNG, the 300 project) in an identical manner
to our standard analyses. As such we shall be able to probe the realism of
hydrodynamical simulations, as well as explore sources of measurement bias in
X-ray observables.

## Future Work

In the future we intend to introduce support for the analysis of X-ray
telescopes other than _XMM_ -Newton, first focusing on _Chandra_ and _eROSITA_
, and then possibly considering the addition of other X-ray instruments. This
will include the same ability to find relevant data and generate data products
as is already implemented for _XMM_ , and will also involve the introduction
of powerful multi-mission joint analyses to fully exploit the X-ray archives.

We have already begun work that pairs XGA with XCS’ simulated XMM observations
(see Figure 3), and planned support for other telescopes will also extend to
the analysis of simulated observations from future telescopes (e.g. ATHENA and
Lynx).

We are also happy to work with others to introduce specific analysis features
that aren’t already included in the module.

## Acknowledgements

David J. Turner (DT), Kathy Romer (KR), and Paul A. Giles (PG) acknowledge
support from the UK Science and Technology Facilities Council via grants
ST/P006760/1 (DT), ST/P000525/1 and ST/T000473/1 (PG, KR).

David J. Turner would like to thank Aswin P. Vijayan, Lucas Porth, and Tim
Lingard for useful discussions during the course of writing this module.

We acknowledge contributions to the XMM Cluster Survey from A. Bermeo, M.
Hilton, P. J. Rooney, S. Bhargava, L. Ebrahimpour, R. G. Mann, M.
Manolopoulou, J. Mayers, E. W. Upsdell, C. Vergara, P. T. P. Viana, R.
Wilkinson, C. A. Collins, R. C. Nichol, J. P. Stott, and others.

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# Slow Inter-area Electro-mechanical Oscillations Revisited:
Structural Property of Complex Multi-area Electric Power Systems

Hiya Akhil Gada1 and Marija D. Ilic2 This material was supported by the MIT
Aero-Astro Distinguished Scholars Program and the NSF Grant No. ECCS-2002570
entitled ASCENT: Boosting Cyber and Physical Resilience of Power Electronics-
Dominated Distribution Grids.The authors also greatly appreciate discussions
with Dr. Rupamathi Jaddivada.1 Hiya Akhil Gada<EMAIL_ADDRESS>is a Ph.D.
candidate at the Laboratory for Information and Decision Systems (LIDS) of
Massachusetts Institute of Technology, Cambridge, MA, USA<EMAIL_ADDRESS>Marija D. Ilic is a Joint EECS Adjunct Professor and a Senior Research
Scientist at Laboratory for Information and Decision Systems (LIDS),
Massachusetts Institute of Technology (MIT), Cambridge, USA. She is also a
Professor Emerita in the Electrical and Computer Engineering (ECE) Department
of Carnegie Mellon University, Pittsburgh, PA<EMAIL_ADDRESS>
###### Abstract

This paper introduces a physically-intuitive notion of inter-area dynamics in
systems comprising multiple interconnected energy conversion modules. The idea
builds on an earlier general approach of setting their structural properties
by modeling internal dynamics in stand-alone modules (components, areas) using
the fundamental conservation laws between energy stored and generated, and
then constraining explicitly their Tellegen’s quantities (power and rate of
change of power). In this paper we derive, by following the same principles, a
transformed state-space model for a general nonlinear system. Using this model
we show the existence of an area-level interaction variable, $intVar$, whose
rate of change depends solely on the area internal power imbalance and is
independent of the model complexity used for representing individual module
dynamics in the area. Given these structural properties of stand-alone
modules, we define in this paper for the first time an inter-area variable as
the difference of power wave incident to tie-line from Area I and the power
reflected into tie-lie from Area II. Notably, these power waves represent the
interaction variables associated with the two respective interconnected areas.
We illustrate these notions using a linearized case of two lossless inter-
connected areas, and show the existence of a new inter-area mode when the
areas get connected. We suggest that lessons learned in this paper open
possibilities for computationally-efficient modeling and control of inter-area
oscillations, and offer further the basis for modeling and control of dynamics
in changing systems comprising faster energy conversion processes.

##  I Introduction

The existence of low-frequency electro-mechanical oscillations has been
recognized and studied as a long-standing industry problem [1, 2], which has
recently become even more pronounced. To mention a few, electric power systems
have experienced these problems in the form of sub-synchronous resonance (SSR)
between turbine modules and series compensated transmission lines [3]; low-
frequency oscillations between different control areas during loss of large
power plants [4]; and, more recently, wide-area oscillations, such as between
Ukraine and Spain, and between Sweden and Spain [5]. The problem of forced
oscillations has also been studied [6].

As these problems continue to emerge, it has become very difficult to identify
their root causes, and to design control needed to suppress them. Most of the
efforts have gone into transient stability simulations and small signal
stability analyses, in particular as the slow power plants get replaced by
smaller-scale Inverter Based Resources (IBRs) [5]. A particular new challenge
perceived by the industry is a lack of detailed models of diverse components.
Also, several renown researchers have proposed in the past modeling and
analyses approaches, notably use of normal forms [7] and notions of
dissipating energy flows [8]. While these methods help with locating the
source of inter-area oscillations, they do not model the inter-area
interactions dynamically.

The primary contribution of this paper is to meet the need to have a
physically-intuitive interpretation of dynamic interactions, in terms of
energy conversion, power and rate of power exchanges, between different energy
conversion components and their relations to the existence of inter-area
oscillations within an interconnected electric energy system. We do this by
observing that each stand-alone module $i$ (component, sub-system) can be
characterized in terms of its own internal states and the shared [9]
interaction variable $intVar^{i}$ which is a function of its own internal
states only, as introduced in our earlier work [10, 11]. We then provide a
notion of dynamic interaction between two areas by introducing, for the first
time, an inter-area variable as the difference of the interaction variables of
the respective areas.

In Section II we review a previously introduced aggregate state space model
representing dynamics of energy and power within a multi-area interconnected
electric power system; we review a previously-introduced structural definition
of a technology-agnostic interaction variable associated with each stand-alone
module $i$; and, define electro-mechanical inter-area oscillation as an
interplay between interaction variables of neighboring modules $j\in C_{i}$.
We stress that these concepts are general and applicable to systems comprising
both conventional generation and renewable resources.

Next, in Section III, we illustrate a particular case of linearized power-
frequency energy dynamics comprising conventional power plants. This section
is a short summary of the earlier work [10] which laid the foundations for
claiming the existence of interaction variables in linearized models of
electric power grids. An important aspect of this modeling of power dynamics
is that the load power deviations from forecast can be explicitly modeled.

Section IV concerns eigenvalue analyses of states contributing to inter-area
oscillations. To start with, the existence of an interaction variable
associated with a stand-alone module is a direct result of its own power
conservation. Importantly, the structural existence of an eigenmode in an
interconnected system, not present in two disconnected subsystems, is now
shown to be a direct consequence of power conservation, both mathematically
and through simulations. Based on these properties, in sub-section IV-A we
propose a method for quickly identifying the modules whose states contribute
the most to the inter-area oscillations. This method could be potentially of
major use in real-world large-scale systems for which understanding causes of
inter-area oscillations becomes a major problem [5].

In Section V we describe numerical simulations to illustrate the sensitivities
of inter-area oscillations with respect to the strength of tie-line flow
connections and inertia of generators. For the first time the effects of grid
design and energy conversion processes on physically-intuitive properties such
as frequency of inter-area oscillations can be interpreted without requiring
full-blown time domain simulations.

In Section VI we add an intermittent source and simulate an example to show
its effect on the inter-area oscillations depending on the frequency of
intermittent disturbances. We close in Section VII by stressing the structural
nature of inter-area oscillations and potential for further generalization of
these properties to multi-energy systems.

##  II  Structural property of complex multi-area electric power systems

In electric power systems, such as the one shown in Figure 1, typically single
port modules, generators and loads, are interconnected via transmission lines
whose real power flows are nonlinearly dependent on the nodal phase angle
differences. All components, except perfect resistors, have state variables
and time constants defining their natural and/or forced response given initial
conditions. In today’s AC electric power systems these state variables are
frequency $\omega(t)$, voltage $V(t)$, real power $P(t)$ and reactive power
$Q(t)$. When writing the dynamic model in terms of these variables, one
quickly loses physical insights. It therefore becomes quite hard to understand
root causes of dynamic problems of interest. To overcome this complexity, we
take a modular approach to modeling complex interconnected electric power
systems.

Figure 1: An interconnected 3-bus 2-area system

We introduce a transformed state space which is derived by combining the
dynamics of conventional internal state variables $\tilde{x}_{old,i}(t)$ in
each module $i$ (except the nodal phase angle $\delta_{i}$) and its generated
power $P_{i}(t)$. The structural form of this transformed state space is
derived by introducing a new, transformed state variable comprising a
technology-dependent $\tilde{x}_{old,i}(t)$ and the power
$P_{i}(t)=\frac{dE_{i}(t)}{dt}$

$\displaystyle
x_{new,i}(t)=\begin{bmatrix}\tilde{x}_{old,i}(t)&P_{i}(t)\end{bmatrix}^{\top}$
(1)

###  II-A General structure of single port dynamic components

We first derive a general structure of single port components by recognizing
that each one can be thought of as an energy conversion processing module
which has certain structural properties resulting from fundamental
conservation of energy.

$\dot{E}_{i}(t)=-\frac{E_{i}(t)}{\tau_{i}}-P_{i}(t)+P_{in,i}(t),\;\;E_{i}(0)=E_{i0}$
(2)

Here $E_{i}(t)$, and $P_{i}(t)$ are stored energy in module $i$ and real power
generated by module $i$ at the port/node. $P_{in,i}(t)$ is the exogenous power
input into module $i$. The term $\frac{E_{i}(t)}{\tau_{i}}$ stands for Joule
losses internal to module $i$, and $\tau_{i}$ is the time constant
representing rate at which energy stored in the module dissipates, and can be
interpreted as the ratio of stored and dissipated energy [12].

We also recall that the rate of change of energy conversion within each module
$\dot{E}_{i}(t)$ is technology-specific and can be expressed in terms of
derivatives of its own local state variables $\dot{\tilde{x}}_{old,i}(t)$. It
then follows from the basic conservation law in (4) when combined with energy
conversion (2) that the dynamics of the local variables $\tilde{x}_{old,i}(t)$
can be expressed structurally in terms of its own states and its rate of
change of energy, namely power.

$\dot{\tilde{x}}_{old,i}(t)=\tilde{f}_{i}(\tilde{x}_{old,i}(t),P_{i}(t),u_{i}(t),m_{i}(t))$
(3)

where $u_{i}(t)$ and $m_{i}(t)$ are the local control and exogenous inputs,
respectively. The key observation when interconnecting modules of diverse
energy conversion components is that power and rate of change of power are
Tellegen’s quantities, as a direct consequence of power conservation [13].
This means that power output from module $i$ must equal power injected into
modules $j\in C_{i}$, where $C_{i}$ is the set of $i$’s neighbouring modules.
For the example of electric power grid, power produced by the generator equals
the sum of power taken out by the components $j$ directly connected to it,
namely transmission lines.

$P_{i}(t)=\sum_{j\in C_{i}}P_{ij}(t)$ (4)

where $P_{ij}(t)$ is the real power flow from module $i$ to $j$. For purposes
of capturing rates of interactions between modules, we observe that rates of
change of power $\dot{P}_{i}(t)$ must balance with the rates of change of
power, say $\dot{P}_{ij}(t)$, taken out by the other components, $j\in C_{i}$,

$\dot{P}_{i}(t)=\sum_{j\in C_{i}}\dot{P}_{ij}(t)$ (5)

The dynamics of $x_{new,i}$ takes on the form,

$\displaystyle\dot{x}_{new,i}(t)$
$\displaystyle=\begin{bmatrix}\dot{\tilde{x}}_{old,i}(t)\\\
\dot{P}_{i}(t)\end{bmatrix}=\begin{bmatrix}\tilde{f}_{i}(\tilde{x}_{old,i}(t),P_{i}(t),u_{i}(t),m_{i}(t)\\\
h_{i}(\tilde{x}_{old,i}(t),\dot{P}_{j},j\in C_{i})\end{bmatrix}$ (6)

###  II-B Existence of interaction variable at component level

Consider a component’s interaction variable as the power generated by the
component $P_{i}$ [10, 11]. By re-writing (2) as

$P_{i}(t)=P_{in,i}(t)-\dot{E}_{i}(t)-\frac{E_{i}(t)}{\tau_{i}}$ (7)

it follows that a disconnected component has $P_{i}(t)=0$ and that it is a
function only of its internal inputs and state variables.

###  II-C Simplified model of two-port components

Similarly, the basic conservation of power for two-port transmission lines
interconnecting its sending end $i$ and receiving end $j$ is

$\frac{dE_{ij}(t)}{dt}=P_{i}(t)+P_{j}(t)-\frac{E_{ij}}{\tau_{ij}}$ (8)

where $\frac{E_{ij}}{\tau_{ij}}$ represents Joule losses in the transmission
line, and, assuming negligible resistance it is approximately zero. When
studying slow electro-mechanical oscillations $\frac{dE_{ij}(t)}{dt}\approx 0$
because the time constants of its LC parameters are near zero relative to the
time constants of electrical machines. Consequently, the line is modeled as
having inductance parameter $X_{ij}$. It follows from (8) that

$P_{i}(t)+P_{j}(t)=0$ (9)

###  II-D Interaction variable of stand-alone area $i$

To review a general structural notion of an interaction variable $intVar^{1}$
of Area 1 proposed in [10, 11], we re-state constraint given in (4) for each
network node within the area by stating a conservation of rate of change of
power for the two modules (generator-load pairs are taken as one module) in
$Area\;1$

$P_{i}=\sum_{j\in C_{i}}P_{ij}(t)$ (10)

For all $i\in Area\;1$ in the network shown in Figure 1 power generated by the
two modules at the two nodes are $P_{1}^{1}(t)=P_{G1}^{1}(t)-P_{L1}(t)$, and
$P_{2}^{1}(t)=P_{G2}^{1}(t)-P_{L2}(t)$, respectively. Further, when applying
constraint given in (4) we get,

$\dot{z}^{1}(t)={intVar}^{1}(t)=P_{1}^{1}(t)+P_{2}^{1}(t)=F_{1-2}$ (11)

where $F_{1-2}$ is the power flow from Area 1 to Area 2 through the tie-line.
We can infer that, for a disconnected area, $F_{1-2}(t)=constant$ and
$\dot{F}_{1-2}(t)=0$, i.e. net power out of disconnected area must remain
constant and its derivative is zero. This linear combination of net power
generated by all modules within a given Area 1 is defined as the
$intVar^{1}(t)$ of Area 1 with the rest of the system [10, 11].

###  II-E Inter-area dynamics: Definition in terms of interaction variables of
stand alone areas

We propose, given the above fundamental relations for energy conversion, and
the notion of stand-alone of interaction variable $intVar^{I}(t)$, for each
area $I$ that when interconnecting the two areas shown in Figure 1 via
transmission line, whose parameter is $X_{2}$, general Tellegen’s Theorem in
(4) implies:

  * •

Incident power into transmission line from Area $1$ must be the same as the
power generated by the Area $1$, given in (7), $I=1$.

  * •

Reflected power into transmission line from Area $2$ is given by the same (7),
$I=2$

  * •

Definition: Inter-area variable I-II is the difference between incident and
reflected powers into the tie line interconnecting them.

It can be seen from (7) that the incident power is mainly contributed by some
internal power injected into Area 1 reduced by the Joule losses inside the
area and the rate for change of energy consumed by the area. These definitions
of incident and reflected power are degenerate since the travelling waves of
tie-line are not modeled when capturing slow electro-mechanical inter-area
dynamics which is mainly created by the difference in powers injected from the
areas.

##  III Linearized modeling of power dynamics

In this section we suggest that the linearized modeling of power dynamics
introduced earlier in [10] follows from the more general derivations in
Section II which do not require linearization and are technology-agnostic. For
completeness, we briefly summarize this early derivation. To do so, we
consider the same two-area interconnected system shown in Figure 1, which
comprises a mix of conventional generators and loads interconnected via
lossless transmission lines.

###  III-A Dynamics of a single generator

The linearised model of a single governor-turbine-generator (G-T-G) set
includes the rotor swing equation, the turbine dynamics and the local governor
feedback proportional control. The standard state space form of a governor-
turbine-generator (G-T-G) (12) has the generator state $\omega_{G}$, the
turbine state $P_{t}$ and the governor state $a$, all being the deviations
from their nominal value.

$\displaystyle\dot{x}_{LC}$ $\displaystyle=\begin{bmatrix}\dot{\omega}_{G}\\\
\dot{P}_{t}\\\
\dot{a}\end{bmatrix}=\begin{bmatrix}\frac{-D}{M}&\frac{1}{M}&0\vspace{0.15cm}\\\
0&\frac{-1}{T_{t}}&\frac{K_{t}}{T_{t}}\vspace{0.15cm}\\\
\frac{-1}{T_{g}}&0&\frac{-1}{rT_{g}}\end{bmatrix}\begin{bmatrix}{\omega}_{G}\\\
{P}_{t}\\\ {a}\end{bmatrix}+\begin{bmatrix}\frac{-1}{M}\\\ 0\\\
0\end{bmatrix}P_{G},$ $\displaystyle=A_{LC}x_{LC}+c_{M}P_{G},$ (12)

where $M,D,T_{t},T_{g}$ are moment of inertia of the combined G-T-G set, its
damping coefficient, and the time constants of the turbine and governor,
respectively. $\omega_{G},P_{G},P_{t},a$ are the rotor frequency of the
generator, its real power output, power produced by the turbine, and the valve
opening, respectively. $K_{T}$ and $r$ are the local control gains in the
turbine and governor respectively. This model is effectively a particular case
of the general nonlinear dynamics of a generator (6) in transformed state
space.

###  III-B Network constraints

Real power flow into a node is a function of the nodal voltage magnitudes and
phase angles of the network, $P^{N}=P^{N}(\delta,V)$. Linearizing power flow
into a generator bus and a load bus under the decoupling assumption ($\partial
P^{N}/\partial V=0$) gives us,

$\displaystyle P_{G}+F_{G}$ $\displaystyle=J_{GG}\delta_{G}+J_{GL}\delta_{L}$
(13) $\displaystyle F_{L}-P_{L}$
$\displaystyle=J_{LG}\delta_{G}+J_{LL}\delta_{L}$ (14)

where the network has $n$ generators and $m$ loads. $\delta_{G}$ and
$\delta_{L}$ correspond to the vector of phase angles, and $F_{G}$ and $F_{L}$
correspond to the real power flow from neighbouring modules at the generator
and load bus respectively. The network constraints given in (13) and (14) are
well-known linearized power flow equations, and represent a particular case of
a more general power balance given as (4) earlier. The main step in deriving a
transformed state space requires applying general constraints on rate of
change of power shown in (5) introduced earlier, implying that differentiation
of power balance equations is a valid step. In today’s power systems problems
$P_{L}$ is predicted and dynamical system response to its deviations
$\dot{P}_{L}$ is of direct interest. This is obtained by expressing
$\omega_{L}$ as a disturbance caused by the load power deviations
$\dot{P}_{L}$ and replacing it in the network constraints imposed on the rate
of change of power generation $\dot{P}_{G}$ In short, define

$\displaystyle
K_{P}=J_{GG}-J_{GL}J_{LL}^{-1}J_{LG},\;\;D_{P}=J_{GG}J_{LL}^{-1}$ (15)

Differentiate and rearrange (13) and (14), to obtain

$\displaystyle\dot{P}_{G}$
$\displaystyle=K_{P}\omega_{G}+\dot{F_{e}}-D_{P}\dot{P}_{L}$ (16)

Here $F_{e}=J_{GG}J_{LL}^{-1}{F}_{L}-F_{G}$ represents the effective flow from
neighbouring areas as seen by each generator. Equations (12) represent
linearized model of G-T-G components in the entire system and when subject to
Tellegen’s constraints on their rate of change power in the interconnected
system, given as (16), jointly result in a particular case of transformed
state space model introduced earlier in (6).

###  III-C Interconnected system dynamics in transformed state space

It was shown in [10, 11] that when using the vector and diagonal matrix
notation for a general power system, and combining the local generator
dynamics and the network constraints, one obtains without loss of generality
transformed state-space model of the system [10, 11]

$\displaystyle\begin{bmatrix}\dot{x}_{LC}\\\
\dot{P}_{G}\end{bmatrix}=\begin{bmatrix}A_{LC}&C_{M}\\\
K_{P}E&0\end{bmatrix}\begin{bmatrix}{x}_{LC}\\\
{P}_{G}\end{bmatrix}+\begin{bmatrix}0\\\
\dot{F}_{e}\end{bmatrix}+\begin{bmatrix}0\\\ -D_{P}\dot{P}_{L}\end{bmatrix}$
(17)

The system matrix of the augmented state space takes on the form

$\displaystyle A=\begin{bmatrix}A_{LC}&C_{M}\\\ K_{P}E&0\end{bmatrix}$ (18)

This model is basically a linearized model of a more general nonlinear model
derived earlier in (17).

##  IV Eigen-mode Analysis of Inter-area Dynamics

In this section, we make use of the transformed state space model shown above
and its system matrix given in (18) to carry our eigen-mode analysis. In
particular, given structural properties of this model and its system matrix,
we wish to assess relations between the eigen-modes in disconnected control
areas as well as in the interconnected system. We expect that each
disconnected area itself has a zero eigen-mode reflecting structural
constraint on its Tellegen’s quantities, power and/or rate of change of power.
These modes are naturally contributed by the states internal to the areas. We
also anticipated that an additional eigen-mode appears when the areas are
interconnected and that the state contributing to this eigen-mode are from
both areas. The inter-area oscillations (IAO) are then contributed by these
states. Using participation factor analysis we identify the state variables of
the components that contribute the most to the IAO eigen-mode. Most
importantly, we illustrate how the transformed state space makes it convenient
to identify causes of oscillations.

###  IV-A Methodology to identify the interconnection eigen-mode

To understand which states contribute the most to the inter-area oscillations,
we first need to identify which eigen-modes of the system arise due to the
interconnection of the areas. A general method to identify the interconnection
eigen-mode is explained below, followed by a simulated example.

For our analysis, consider an undamped single isolated system ($F_{e}=0$) with
constant power load ($\dot{P}_{L}=0$). Its state space formulation reduces to,

$\displaystyle\dot{x}=Ax$ (19)

We simulate two cases,

  * •

Connected Isolated System (CIS): The original system of interest, like in
Figure 1

  * •

Disconnected Isolated System (DIS): The same system in Figure 1 but with a
disconnected inter-area tie line

The structure of the eigen-values of $A$ [10, 11] for DIS is,

$\displaystyle\lambda(A_{DIS})=\begin{bmatrix}0\\\ \vdots\\\ 0\\\ \vdots\\\
\pm\Omega_{1}j\end{bmatrix}$ (20)

where $\pm\Omega_{1}j$ corresponds to the local eigen-values of the two areas.
The zero eigen-values are due the conservation of power/rate of change of
power in each of the areas. The structure of the eigen-values of $A$ [10, 11]
for CIS is,

$\displaystyle\lambda(A_{CIS})=\begin{bmatrix}0\\\ \vdots\\\ \pm\omega j\\\
\vdots\\\ \pm\Omega_{2}j\end{bmatrix}$ (21)

where, like in DIS, $\pm\Omega_{2}j$ corresponds to the local eigen-values of
the two disconnected areas. We have an additional non-zero eigen-value
$\pm\omega j$ which arises due to the interconnection of the two areas and is
responsible for the IAO. To identify the eigen-value that corresponds to the
interconnection, one can compare $\lambda_{DIS}$ and $\lambda_{CIS}$ and find
the additional non-zero interconnection eigen-value $\pm\omega j$. In most
cases, the interconnection eigen-mode will be of a lower frequency than the
local eigen-modes. A simple participation factor analysis can then be done to
find the state variables contributing to $\pm\omega j$.

##  V Sensitivity of inter-area oscillations with respect to system parameters

We simulate an example to better understand the effect of weak inter-area tie
lines and low inertia generators on inter-area oscillations (IAO). Consider
the three-bus two-area system in Figure 1. The system is undamped, isolated
and is subjected to constant power loads $L_{1},L_{2},$ and $L_{3}$. The lines
are taken as purely inductive with reactances $X_{1}$ and $X_{2}$. A similar
analysis can be done for a damped system, but the idea remains the same.

###  V-A Effect of tie-line strength

We demonstrate the effect of the strength of the inter-area tie-line on the
interconnection eigen-mode and IAO by considering three cases.

  1. 1.

Case 1: We keep the reactance of tie-line relatively small and comparable to
the local line strengths.

$X_{2}=X_{1}$

  2. 2.

Case 2: We keep the reactance of tie-line relatively bigger than the local
line strengths.

$X_{2}=10X_{1}$

  3. 3.

Case 3: This is the DIS case, in Figure 1, where the tie-line is very weak and
almost non-existent.

$X_{2}>>X_{1}$

In our eigen-mode and participation factor analysis for case 1, we identify
the following non-zero eigenvalues and their primary contributing state
variables.

$\displaystyle\Omega^{1}$ $\displaystyle=\pm
3.7499j\longrightarrow\omega_{G}^{2},P_{G}^{2}$ $\displaystyle\omega^{1}$
$\displaystyle=\pm
2.1650j\longrightarrow\omega_{G}^{1},P_{G}^{1},\omega_{G}^{3},P_{G}^{3}$

Similarly for case 2,

$\displaystyle\Omega^{2}$ $\displaystyle=\pm
3.1028j\longrightarrow\omega_{G}^{1},P_{G}^{1},\omega_{G}^{2},P_{G}^{2}$
$\displaystyle\omega^{2}$ $\displaystyle=\pm
0.8274j\longrightarrow\omega_{G}^{1},P_{G}^{1},\omega_{G}^{3},P_{G}^{3}$

And for case 3,

$\displaystyle\Omega^{3}$ $\displaystyle=\pm
3.0618j\longrightarrow\omega_{G}^{1},P_{G}^{1},\omega_{G}^{2},P_{G}^{2}$

It is straightforward to identify the interconnection eigen-mode for case 1
and case 2 as $\omega^{1}$ and $\omega^{2}$ respectively. The states of
generator $G_{1}$ and $G_{3}$ contribute to the interconnection eigen-mode,
the mode that disappears in the DIS case 3. We also note that the local eigen-
modes, $\Omega^{1},\Omega^{2},\Omega^{3}$, are primarily dependent on the
states of generators in their respective area.

As defined in Section II, inter-area oscillations are expressed as the
difference between the rate of change of interaction variables of two
disconnected areas. We plot this for the three cases in Figure 2. The weaker
the line gets, the slower is the oscillation of the power flowing through it,
with the oscillation going to 0 for DIS. This can also be seen in the
eigenvalues obtained for each case.

Figure 2: Inter-area oscillation as a difference of the rate of change
interaction variables in the two areas for varying tie-line strength

###  V-B Effect of inertia of generator

We now demonstrate the effect of generator inertia on the interconnection
eigen-mode and IAO. Again, consider the isolated system in Figure 1 and take
two cases.

  1. 1.

Case 1: The inertia of all generators is equal

$M_{G1}=M_{G2}=M_{G3}$

  2. 2.

Case 2: The inertia of generator in area 2 is taken to be relatively bigger
than the other generators

$M_{G1}=M_{G2}<M_{G3}$

We take $X_{1}=X_{2}$ for this particular experiment. The parameters chosen
for our simulation are given in the appendix. On doing eigen-mode analysis and
participation factor analysis for case 1 we get,

$\displaystyle\Omega^{1}$ $\displaystyle=\pm
3.7499j\longrightarrow\omega^{2}_{G},P_{G}^{2}$ $\displaystyle\omega^{1}$
$\displaystyle=\pm
2.1650j\longrightarrow\omega^{1}_{G},P_{G}^{1},\omega^{3}_{G},P_{G}^{3}$

Similarly for case 2,

$\displaystyle\Omega^{2}$ $\displaystyle=\pm
3.5221j\longrightarrow\omega^{1}_{G},P_{G}^{1},\omega^{2}_{G},P_{G}^{2}$
$\displaystyle\omega^{2}$ $\displaystyle=\pm
1.4578j\longrightarrow\omega^{1}_{G},P_{G}^{1},\omega^{2}_{G},P_{G}^{2},\omega^{3}_{G},P_{G}^{3}$

When $G_{3}$ has a large inertia, the interconnection eigen-mode
($\omega^{2}$) gets slower and $G_{3}$ states start contributing more to it.
As is expected, generators with large inertia have a slower response and can
be seen in the Figure 3 which plots the inter-area oscillation for the two
cases.

Figure 3: Inter-area oscillation as a difference of rate of change of
interaction variables from two areas for varying generator inertia

##  VI Impact of Intermittent Resources on IAO

The transformed state space has another advantage; it allows us to express
load disturbances in the form of $\dot{P}_{L}$ (17). Specifically, renewable
sources are intermittent and can be characterised as a varying negative load.
Consider the network in Figure 1, with and without a renewable source in the
place of $L_{1}$. When the damping is insufficient and the renewable power
output deviations include the interconnection mode
($\dot{P}_{L}=A_{\circ}\sin(\omega t)$, $\omega$ is the interconnection
eigenvalue), it resonates with the IAO and makes it unstable as can be seen in
Figure 4.

Figure 4: Inter-area oscillation expressed as a difference of rate of change
interaction variable of the two areas with and without a varying renewable
source

##  VII Concluding remarks

The notion of $intVar$ used in this paper has two unique properties: when the
area is disconnected $intVar$ remains constant, and, its rate of power
exchange with other areas is zero. Given these properties, we claim that the
energy dynamics interpretation of interactions within a multi-area system is
structural, i.e., they depend only on the basic conservation laws and
Tellegen’s theorem and are agnostic to technology of the internal energy
conversion dynamics of modules. This means that the inter-area dynamics only
depend on the range of interface specifications, an essential property for
distributed modeling, sensing, computing and control. An example of aggregate
mathematical model is shown on a small electric power system comprising two
areas to illustrate these structural properties. Future work include,
incorporating real-reactive power coupling and losses in the modeling,
analysing practical examples and studying the relationship between feedback
linearizing control and the structural inter-area property for suppressing
slow inter-area oscillations between New York and New England areas [4] and
for eliminating sub-synchronous resonance [14]. For Section V-A, all
generators are taken to be identical, i.e., their $A_{LC}$ matrix is the same.
The generator inertias are $M=3.2\;\text{p.u.}$ and line reactance
$X_{1}=6.67\cdot 10^{-2}\;\text{p.u.}$. For Section V-B,
$X_{1}=X_{2}=6.67\cdot 10^{-2}\;\text{p.u.}$. For case 1, generator inertias
are $M_{G1}=M_{G2}=M_{G3}=3.2\;\text{p.u.}$ and for case 2,
$M_{G3}=32\;\text{p.u.}$. All simulations show deviations of the quantities
around equilibrium. As both the examples consider an undamped system, $D=0$
and the time constants $T_{t},T_{g}$ are very large numbers.

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