Energy markets and responsive grids modeling, control, and optimization

The IMA Volumes in Mathematics and its Applications

Sean Meyn
Tariq Samad · Ian Hiskens
Jakob Stoustrup  Editors

Energy Markets
and Responsive
Grids
Modeling, Control, and Optimization


The IMA Volumes in Mathematics and its
Applications
Volume 162

Series editor
Daniel Spirn, University of Minnesota, MN, USA


Institute for Mathematics and
its Applications (IMA)
The Institute for Mathematics and its Applications (IMA) was established in
1982 as a result of a National Science Foundation competition. The mission of
the IMA is to connect scientists, engineers, and mathematicians in order to address
scientific and technological challenges in a collaborative, engaging environment,
developing transformative, new mathematics and exploring its applications, while
training the next generation of researchers and educators. To this end the IMA
organizes a wide variety of programs, ranging from short intense workshops in areas
of exceptional interest and opportunity to extensive thematic programs lasting nine
months. The IMA Volumes are used to disseminate results of these programs to the

broader scientific community.
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More information about this series at />

Sean Meyn • Tariq Samad • Ian Hiskens
Jakob Stoustrup
Editors

Energy Markets and
Responsive Grids
Modeling, Control, and Optimization

123


Editors
Sean Meyn
Department of Electrical
and Computer Engineering
University of Florida
Gainesville, FL, USA
Ian Hiskens
Department of Electrical Engineering
and Computer Science
University of Michigan

Ann Arbor, MI, USA

Tariq Samad
Technological Leadership Institute
University of Minnesota
Minneapolis, MN, USA
Jakob Stoustrup
Department of Electronic Systems
Aalborg University
Aalborg, Denmark

ISSN 0940-6573
ISSN 2198-3224 (electronic)
The IMA Volumes in Mathematics and its Applications
ISBN 978-1-4939-7821-2
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Foreword

This volume contains a selection of essays based on a workshop “Control at Large
Scales: Energy Markets and Responsive Grids” held at the Institute for Mathematics
and its Applications from May 9–13, 2016 and organized by Sonja Glavaski, Ian
Hiskens, Sean Meyn, Tariq Samad, and Jakob Stoustrup. These papers provide a
landscape of the mathematical, financial and policy challenges that are present with
the design of an efficient, stable and resilient electrical grid. The workshop ran as
part of an annual thematic year organized by Fariba Fahroo, Tryphon Georgiou,
J.W. Helton, Anders Rantzer, Tariq Samad, Eduardo Sontag and Allen Tannenbaum
on Control Theory and its Applications that ran at the IMA during the 2015–2016
academic year. We would like to especially thank volume editors Ian Hisken, Sean
Meyn, Tariq Samad and Jakob Stroustrup. Finally, we acknowledge the National
Science Foundation for its support of the IMA.
Minneapolis, MN, USA

Daniel Spirn

v


Introduction

The electric power infrastructure in any large region amounts to a system of
systems—dynamically interconnected domains with communication, computation,
and control functions at multiple temporal and spatial scales. The control loops
that regulate electricity exist alongside electricity markets that introduce their own
dynamics as they encourage generators to come on-line, or take a break from
operations. The grid today is remarkably reliable, given its inherent complexity and
uncertainty.
However, a tremendous transformation of the power grid is under way across
the globe. The movement towards a so-called smart grid has been driven by many
different players in industry and by societal pressure—people are concerned about
the future of the planet, and in particular the impact of global warming. A truly smart
transformation of the grid will bring about many societal benefits, including a reduction in pollution and greenhouse gases, reduced capital and operational expenses,
and improved energy security. To ensure that our electricity supply remains reliable
requires careful consideration of control strategies, communications, and market
design.
In the future, as is true today, the ultimate challenge is to control generation,
transmission, distribution, storage, and consumption of electricity. Consumers,
markets, and regulators are also participants and stakeholders, and the multiple roles
and interrelationships may exacerbate the challenge in the absence of appropriate
market rules and control designs. Quoting one of the closing statements of the
first chapter: In order to sustain such a drastic and rapid change, new control
paradigms have to be developed moving the grid to a flexible, cooperative structure
providing survivability of the system. This cannot be achieved without revisiting
traditional reliability criteria and adding such new concepts as resilience, robustness
and flexibility.
The editors of this volume organized the IMA workshop on Control at Large
Scales: Energy Markets and Responsive Grids in May, 2016, as part of the yearlong IMA program on Control Theory and its Applications, held at the University of
Minnesota. The goal of the workshop was to bring together experts and newcomers
interested in all aspects of the challenges facing the creation of a more sustainable

vii


viii

Introduction

electricity infrastructure. Included in the meeting were experts in distributed control,
stochastic control, stability theory, economics, policy, and financial mathematics, as
well as in all aspects of power system operation.
This monograph consists of selected essays by participants in the workshop on
the challenges we face today and in the future, along with potential solutions. All
contributions were subjected to a peer-review process, with significant revisions in
many cases.
The chapters are loosely organized according to theme, beginning with a survey
from three authors from ISO New England. The next few chapters consider several
significant challenges in the domain of market design. A theme in these chapters
is the question of incentives for innovation in markets with significant risk on
many time scales, and where assets may cost billions of dollars. These chapters
are followed by chapters on optimization and distributed control, and the book
concludes with articles addressing resilience and vulnerability.
Large-scale renewable generation, distributed energy resources, integration of
supply-side and demand-side management, and dynamic markets herald a revolutionary change in power systems. The associated challenges are daunting and will
require multidisciplinary approaches. With the breadth and depth of expertise it
encapsulates, we are hopeful that this volume will contribute towards the envisioned
future for serving humanity’s energy needs.
We are grateful to our authors for their patience with the review process and other,
less excusable, delays. The workshop itself was a hive of discussion and debate
and all participants deserve our thanks as well. As with all IMA workshops, the
arrangements were excellent and allowed the organizers to dedicate their attention

to the workshop technical program. We would like to thank Fadil Santosa, the IMA
Director, in particular for his support and encouragement. Finally, it has been a
pleasure to work with the Springer team: Achi Dosanjh, Nick Valente, and Danielle
Walker.
Gainesville, FL, USA
Minneapolis, MN, USA
Ann Arbor, MI, USA
Aalborg, Denmark

Sean Meyn
Tariq Samad
Ian Hiskens
Jakob Stoustrup


Contents

How to Manage the Complexity of the Grid? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eugene Litvinov, Feng Zhao, and Tongxin Zheng

1

Naïve Electricity Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David B. Spence

29

Capacity Markets: Rationale, Designs, and Trade-Offs . . . . . . . . . . . . . . . . . . . . .
Alfredo Garcia

59

Redesign of US Electricity Capacity Markets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Robert W. Moye and Sean P. Meyn

73

A Swing-Contract Market Design for Flexible Service Provision
in Electric Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Wanning Li and Leigh Tesfatsion
A Dynamic Framework for Electricity Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Anuradha Annaswamy and Stefanos Baros
Fast Market Clearing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Arvind U. Raghunathan, Frank E. Curtis, Yusuke Takaguchi,
and Hiroyuki Hashimoto
Small Resource Integration Challenges for Large-Scale SCUC . . . . . . . . . . . . 177
Cuong Nguyen, Lei Wu, Muhammad Marwali, and Rana Mukerji
Multi-Grid Schemes for Multi-Scale Coordination of Energy Systems . . . . 195
Sungho Shin and Victor M. Zavala
Graphical Models and Belief Propagation Hierarchy
for Physics-Constrained Network Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Michael Chertkov, Sidhant Misra, Marc Vuffray,
Dvijotham Krishnamurthy, and Pascal Van Hentenryck
Profit Maximizing Storage Integration in AC Power Networks . . . . . . . . . . . . 251
Anya Castillo and Dennice F. Gayme
ix


x

Contents

Virtual Inertia Placement in Electric Power Grids. . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Bala Kameshwar Poolla, Dominic Groß, Theodor Borsche,
Saverio Bolognani, and Florian Dörfler
A Hierarchy of Models for Inverter-Based Microgrids . . . . . . . . . . . . . . . . . . . . . . 307
Olaoluwapo Ajala, Alejandro D. Domínguez-García, and Peter W. Sauer
Asynchronous Coordination of Distributed Energy Resources
with Packetized Energy Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
Mads Almassalkhi, Luis Duffaut Espinosa, Paul D. H. Hines, Jeff Frolik,
Sumit Paudyal, and Mahraz Amini
Ensemble Control of Cycling Energy Loads: Markov Decision
Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Michael Chertkov, Vladimir Y. Chernyak, and Deepjyoti Deka
Distributed Control Design for Balancing the Grid Using
Flexible Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Yue Chen, Md Umar Hashmi, Joel Mathias, Ana Buši´c, and Sean Meyn
Disaggregating Load by Type from Distribution System
Measurements in Real Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
Gregory S. Ledva, Zhe Du, Laura Balzano, and Johanna L. Mathieu
Risk-Aware Demand Management of Aggregators Participating
in Energy Programs with Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
William D. Heavlin, Ana Radovanovi´c, Varun Gupta, and Seungil You
Toward Resilience-Aware Resource Allocation and Dispatch
in Electricity Distribution Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
Devendra Shelar, Saurabh Amin, and Ian Hiskens
A Cautionary Tale: On the Effectiveness of Inertia-Emulating Load
as a Cyber-Physical Attack Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Hilary E. Brown and Christopher L. DeMarco

How to Manage the Complexity of the
Grid?
Eugene Litvinov, Feng Zhao, and Tongxin Zheng

“. . . complex systems are counterintuitive. That is,
they give indications that suggest corrective action
which will often be ineffective or even adverse in its results.”
Forrester, Jay Wright

Abstract Power industry is facing revolutionary changes. The direction of the
US Government to low carbon footprint and, as a consequence, high penetration
of renewable energy resources and smart grid technologies are completely transforming planning and operational patterns for electric grid. As more and more
variable and demand response resources being integrated into the electric grid,
the grid operation is experiencing increasing level of uncertainties. The decisionmaking process under such environment becomes more challenging. The grid
architecture and control also become more and more decentralized requiring new
control paradigms and reliability metrics to be investigated in order to achieve much
higher level of flexibility and resilience. These changes are disruptive enough to
cause even transformations in utility business dealing with completely unknown
situations. On the other hand, the evolution in computing; generation, transmission,
and distribution technologies; and mathematical methods creates opportunities for
innovation in power system design and control. New mathematical models for power
system analysis and operation are being developed to address above challenges. We
will discuss the need for new power system control and electricity market design
directions while managing grid complexity.

E. Litvinov ( ) · F. Zhao · T. Zheng
ISO New England Inc., Holyoke, MA, USA
e-mail: ; ;
© Springer Science+Business Media, LLC, part of Springer Nature 2018

S. Meyn et al. (eds.), Energy Markets and Responsive Grids, The IMA Volumes
in Mathematics and its Applications 162,
/>
1


2

E. Litvinov et al.

1 Electric Grid Architecture Evolution
Modern power systems are going through different stages of evolution driven by
technical, economic, and regulatory events. They went from decentralized, very
loosely coupled grid to highly interconnected and centrally controlled systems.
The increased complexity and lack of ability to manage it led to major blackouts
forcing significant changes in system planning and operation. The Great Northeast
Blackout of 1965 led to the creation of the power pools with control centers running
energy management systems (EMS) and centralized regional planning and control.
Each pool linked together multiple neighboring transmission companies with much
stronger ties among them (Figure 1). Besides local control centers, power pools
created pool control centers. Not only did this help in increasing reliability and
resilience by the ability to provide balancing assistance, but also created savings for
the member companies by using less expensive generation to meet the regional load.
The interties between the pools were still weak and only used for emergency help.
With the inception of the markets in the late 1990s and the creation of ISOs/RTOs,
market players started placing economic transactions across the pool boundaries,
increasing the complexity of the grid operation. This led to the reinforcement
of the transmission system and tighter integration of the interconnected systems.
The complexity of such an architecture required new ways of system control. The
economic dispatch (ED) being done in each market area independently created socalled seams issues – inefficient utilization of the interties. This, in turn, requires

additional information technology and communication infrastructure to coordinate
market operations across large geographic areas. The electric grid had become
a very large complex cyber-physical system. All these changes and attempts to
increase grid reliability have not lowered the risk of large blackouts. On the contrary,
the number and frequency of blackouts are increasing, which is the property of a
very large complex system that exhibits self-organized criticality [1]. The blackouts
follow the power law.
Currently, the power industry is facing another revolutionary change. Government directives to lower the carbon footprint and, as a consequence, high
penetration of renewable energy resources and smart grid technologies are completely transforming planning and operational patterns of the electric grid again.

CA2

TO1

PCC

TO3

TO2

CA1

CA3

Fig. 1 Creation of power pools

CA


How to Manage the Complexity of the Grid?

3

Fig. 2 Proliferation of DER

Distributed energy resources are being built deeply in the distribution networks, and
the boundary between transmission, sub-transmission, and distribution is blurring.
Traditionally, electric grid upgrades have been done centrally during transmission
planning process. The process follows very strict reliability standards and requires
large number of system studies, both in the steady state and transient regimes. Today,
numerous changes to the grid are made ad hoc: distributed generation, microgrids,
storage, etc. System operators lose control of the network perimeter. That topological uncertainty adds to the intermittent nature of the renewable resources. The
architecture of the modern grid becomes more and more decentralized, while the
control architecture is staying the same (Figure 2). Significant part of the generation
resources is unobservable to the system operators. The unprecedented level of
uncertainty is introduced not only in the location of distributed resources but their
intermittent nature as well. The output of wind and PV generation can also swing
significantly in time. The tribal knowledge of system operators is failing in dealing
with completely different patterns of the system behavior. Even the concept of
contingency is changing from being binary (the element of the grid is on or off) to
continuous in time. The system load or generation can change by several gigawatts
in a comparatively short period of time. This behavior, considered as abnormal
or emergency, becomes part of the normal operation. This creates tremendous
complexity in power system control.
In addition to DER proliferation, new “green” policies and low gas prices are
causing retirements of coal, oil, and nuclear stations which leads to significant
change in the generation mix and even capacity shortage. This as well makes


4

E. Litvinov et al.

real-time operation decision-making process much more complicated and counterintuitive. Implementation of green and smart grid technologies is significantly
increasing amount of power electronics connected to the transmission and distribution networks. Interactions of such a large number of interconnected controllers
introduce another level of complexity and potential stability problems.
Another property of large cyber-physical systems is high interdependence of
different infrastructures. Not only do we have to monitor electric grid contingencies
but the failures in communication and information technology systems as well. The
system resilience is getting much weaker, which requires new solutions for system
planning and operation. Today, power systems are operated almost exclusively under
the preventive paradigm. Every contingency is considered to be of probability 1, and
the system is dispatched in such a way that no one failure would cause the violation
of reliability criteria (N-1 standard). This approach, being quite expensive in the first
place, becomes economically prohibitive in the new environment. More corrective
actions must be introduced to make power system operation less expensive.
In order to understand the change in the power system operation, one can use DyLiacco’s system state diagram [2] as shown in Figure 3. Each state is triggered by
certain events and characterized by either getting very close to or violating specific
constraints: physical, reliability, economic, etc. In the “alert” state system operator
is facing a trade-off between preventive and corrective actions. By using preventive
actions, the operator forces system away from the operating constraints increasing
the margins. Alternatively, he/she may decide to defer actions until the system enters
into the “emergency” state, especially if the process of moving from “alert” to
“emergency” is comparatively slow. This is definitely a choice between reliability
and economics. In the system with a reasonable level of uncertainty, the operator’s
actions are comparatively stable under wide range of conditions and situations. With
the introduction of much higher level of uncertainty, the conditions that traditionally

Fig. 3 New state transition diagram

How to Manage the Complexity of the Grid?

5

are considered being “alert” become everyday “normal” phenomena, so we are
observing the merging of these two states (Figure 3). Under new circumstances, the
economics of the trade-off between preventive and corrective actions is changing.
Corrective actions and remedial action systems (RAS) become more economic to
use, which, in turn, forces the industry to review its control paradigm.
The complexity induced by the large-scale distributed components, the
lack of observability, and the uncertainty in the future grid brings significant
challenges in modeling, decision-making, and control of the system. To manage
the above complexity by addressing these challenges, the industry needs different
control paradigm, new grid architecture, new algorithms, new models, and new
reliability criteria. The foundation for these new changes should be a more flexible
grid architecture, e.g., a decentralized and distributed grid. Decision-making for the
grid will have to be augmented by lowering the interdependence among different
components and using robust solutions that are insensitive to external disturbances
and economically efficient at the same time. The resulting robust components
in turn will enable flexibility in distributed control structures and achieve the
increasingly needed resilience of the grid. To efficiently design and implement such
control architecture, we will need to formalize the new concepts of resilience and
survivability and create metrics to be used to manage quality of the control.
In the following, we first discuss the general needs for control architecture
(Section 2) and the likely additional control components needed for the existing
control centers (Section 3). Then we explore some specific aspects of the new control
architecture: the corrective controls (Section 4), the uncertainty management
(Section 5), the system flexibility (Section 6), the coordination algorithm (Section 7), and the new system resilience metrics (Section 8). These aspects are by no
means the complete list, but rather reflect what we have considered some major new

pieces that will be needed for a future grid control.

2 New Power System Control Architecture
The new grid needs more flexibility to be able to operate with so much uncertainty.
The flexibility is a very fuzzy concept and being used very loosely in the industry.
It has to be formalized to be used in control and design algorithms. An attempt of
such formalization is presented later in this chapter.
The industry is also very imprecise about the control architecture of the
grid. Many different definitions of the control architecture being used: centralized/decentralized, hierarchical, coordinated, hierarchical-coordinated, distributed,
collaborative, cooperative, etc. All these terms are not clearly defined even in the
control theory literature and, in our opinion, require special attention from the
control community. Today’s control seems to be strictly hierarchical and centralized.
Such system is very rigid and has very little room for flexibility. With the increasing
complexity, such an approach is insufficient to maintain system reliability and
resilience.


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E. Litvinov et al.

Changing only the grid architecture to provide more flexibility while maintaining
reliability is not sufficient. In order to reduce complexity, we have to make control
system flexible as well, with the ability to adapt to different system states. This is
impossible without some degree of distributed decision-making and decentralized
control adapting to the unknown and dynamic environment. Additionally, decentralized systems are more resilient to disturbances or faults. These new qualities
could be achieved by implementing distributed cooperative control paradigm with
the capability of assembling temporary control entities collaborating in addressing
specific events. Such a capability would allow decomposing a very complicated
control problem into smaller, more manageable tasks. Large percentage of the

system events are developing slowly enough so the corrective control would be
capable of addressing large number of events. A new generation of state monitoring
systems should be developed to take advantage of new information available
from different devices and sensors. Decentralized control also requires careful
design of the standard communication and control protocols and interfaces to
enable interaction among heterogeneous components while cooperating in solving
a common problem.
The increase of the computational capabilities and new IT architectures create
opportunities for implementation of innovative control algorithms and infrastructure. Rapidly evolving cloud technology introduces unprecedented capabilities for
online cooperation and collaboration. Being accessible from geographically wide
area and capable of high-performance computing, cloud could serve as a medium
for decentralized and distributed decision-making and control. The tremendous
flexibility of this computing infrastructure will very quickly transition from very
simple to highly complex control problems as needed. A simple example of such
problem is resolving anticipated imbalance caused by a major contingency with the
help of neighboring systems:
• Assembling model on the fly.
• Communicating coordination constraints (max imbalance allowed by participating entities), etc.
• Once resolved, the temporary collaborator is dropped.
Another benefit is ability to capture, accumulate, and use the patterns of the best
control actions and strategies making it available during future events – stigmergy
[3]. The system of such complexity also requires a different approach to reliability.
Being under stress most of the time, power grid has to develop a survivability
property, which is more general than just reliability. In addition, new reliability
criteria together with resilience have to be investigated and implemented in order
to formalize the objective of the power system control and required constraints.


How to Manage the Complexity of the Grid?

7

3 Introducing New System Components to Control Center
The majority of power systems in the US are operated in an organized market
environment or controlled by the RTO/ISO. In general, RTO/ISO performs two
major functions: maintaining reliable system operation and managing wholesale
electricity markets. Both functions can be considered as centralized control.
Modern power system operation deals with the physical aspect of the electric grid, and it is a challenging task. It involves many interacting processes.
These processes can start from planning the system operating mode, coordinating
generation and transmission outages with market participants and local control
centers, forecasting system conditions, committing units for the real-time operation,
scheduling generator outputs and interchanges with external control areas to meet
the varying demand, collecting real-time system operating information through
the supervisory control and data acquisition (SCADA) system, monitoring and
alleviating static and dynamic security violations in the transmission system,
and maintaining system voltages and frequency through the automatic generation
control to taking emergency actions such as demand response, load shedding,
emergency purchases, as well as conducting system restoration after a blackout.
Some of these processes are automatic, and some of them require operators’ manual
actions.
Market operations, on the other hand, deal with the financial aspect of the electric
system. Depending upon the structure of each regional market, each RTO/ISO
may have different market operation procedures. However, broadly speaking, it
includes clearing and settling the day-ahead energy, real-time energy, ancillary
service markets, financial transmission rights (FTR), and forward capacity markets,
monitoring and mitigating market power, and assessing the financial risk of market
participants. Market operation and system operation are interconnected and affect
each other. This is especially true for the real-time market and due to the fact that
financial markets consider physical limitations of the transmission system.
The current RTO control system can be divided into two subsystems, the market

system (MS) and the EMS, as shown in the dotted region of Figure 4. The
market system performs all the market operation functions as described above,
and the EMS facilitates the execution of all system operation processes. With
the increasing penetration of renewables, distributed energy resources, demand
response, and grid level smart devices, system operators are facing a much more
complex system that contains a large number of controllable transmission and
generation resources, various control models, fast-changing operating conditions,
a high degree of uncertainty and is vulnerable to the changes in such external
systems as the fuel delivery system, the regulatory regime, and commodity and
financial markets. The existing control structure needs to be enhanced to facilitate
the management of ever increasing complexity. In Figure 4, three new subsystems
are introduced: dynamic decision support system (DDSS), risk management system
(RMS), and market analysis, training, and simulation system (MATSS).


8

E. Litvinov et al.

Fig. 4 System components for the future RTO

DDSS is a system that provides valuable control parameters to the system and
market operation. The system is dynamic in the sense that it utilizes the latest
available information in producing operational parameters. DDSS may have many
functions and utilize different technologies depending on the task at hand. It should
have the capability to perform the day-ahead and real-time renewable forecast
including wind, solar, and DERs. It provides system operator with the most recent
state of the system. Wide area monitoring using the phasor measurement unit
(PMU) technology is a perfect fit to this task. Online dynamic security analysis
or cascading event analysis will help the system operator define the secure region

of the current system and provide possible corrective action plans. Online interface
limit calculation and adaptive line rating [4] are also key functions of DDSS.
RMS is a system that deals with the increasing level of uncertainty faced by
RTOs. It contains three major functions: collecting statistical information, assessing
the system risk, and mitigating risk. Historical data, such as area control errors, load,
wind production, solar generation, interchange level, transmission and generation
failures, gas pipeline capacity reductions, etc., can be collected for statistical analysis. The system risk can then be assessed based on the statistical model established
using historical data. Different risk indices, such as operational flexibility index
[5], static security severity index [6], short-term loss of load expectation, etc.,
can be computed and displayed to the system operator. Different risk management
techniques can be used to mitigate the system risk. They include, but not limited
to, stochastic [7] and robust unit commitment [8], risk-based economic dispatch [9],
dispatch with ramp constraints [10], etc.


How to Manage the Complexity of the Grid?

9

MATSS performs an important function in assessing the efficiency of both
market and system operations. As a recent trend, market operation is tightly
integrated with the system operation. Actions taken in the system operation could
have a large financial impact on the market participants. A comprehensive market
simulator that is integrated with the traditional dispatcher training system is a very
useful tool in simulating different system and market conditions, quantifying the
financial impact of operator actions, and measuring the operational efficiency. In
addition, such a simulation environment can be used to test future market designs,
to assess the market competitiveness, and to perform the cost-benefit analysis of
new market designs.
DDSS, RMS, and MATSS interact with MS and EMS directly and provide

valuable information such as risk index, system security, cost of actions, and
corrective action plans to the system operators. Introducing three new subsystems
into the existing control scheme could help the system operator to better manage the
increased complexity of the power system.

4 Exploring Corrective Controls
Under today’s centralized control scheme, the risk associated with the power system
uncertainty is mostly managed through preventive actions by the system operator. A
typical example is the enforcement of contingency power flow limits. Namely, the
power flow under any contingency will be within the safety limits, e.g., long-term
emergency (LTE) limits, even without any remedial actions. However, in reality,
a power line has different ratings such as short-term emergency (STE) and LTE,
each associated with certain sustainable time based on thermal conditions. An STE
rating associated with a short time period is higher than an LTE rating associated
with a longer time period, indicating that the line can sustain a higher power level
for a shorter time period. This feature could allow the contingency power flow to
go above the current LTE limit without causing system reliability issues, provided
that corrective actions such as unit redispatch can be taken to return the flow
back to LTE within a certain time period. Consideration of such post-contingency
corrective actions in the dispatch problem allows additional choices, thus providing
more flexibility for the system control and lowering the dispatch cost [11–14].
With increasing penetration of renewable resources, such flexibility becomes more
important because the conventional “preventive” control that requires covering
every possible contingency scenario without factoring in the available corrective
actions would become prohibitively expensive and may even lead to infeasibility.
Below we present mathematical models of how to incorporate corrective actions
into system operator’s dispatch problem.
First consider a conventional security-constrained economic dispatch (SCED)
problem:
minp eT · C(p) , s.t.

(1)


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h(p, d) = 0 ,

(2)

f(p, d) ≤ fmax ,

(3)

fc (p, d) ≤ LTE , ∀ Contingency c

(4)

where p is the vector of unit dispatch decisions, C(p) is the vector of unit dispatch
costs, d is the vector of load at different buses, h() is the power balance equation,
f() is the vector of power flows in monitored lines, fc () is the vector of power flows
under Contingency c, and fmax is the vector of normal ratings of lines.
In the above SCED problem, dispatch decisions p are made such that the power
flow under any contingency would be retained within the safe limit of LTE (4).
This is a very conservative control approach in the sense that the post-contingency
flow could have been allowed to rise above LTE limits for a short time period
(e.g., 15 minutes) without causing network reliability problems. As a result, the
conventional SCED may unnecessarily use some expensive resources to contain a

contingency flow to LTE, despite the chance of that contingency happening could
be slim. With the increasing level of uncertainty in the system, the contingency
definition must be expanded to cover a wide range of uncertainty spectrum, making
the dispatch even more costly. Moreover, the risk of having no dispatch solution
to cover a wide range of contingencies will increase. To address these problems,
considering available corrective actions (e.g., unit redispatch) during contingency
period becomes a natural choice to exploit system flexibility.
The SCED problem with corrective actions can be formulated as the following:
minp,{pc } eT · C(p) , s.t.

(5)

h(p, d) = 0 ,

(6)

f(p, d) ≤ fmax ,

(7)

≤ STE , ∀ Contingency c

(8)

fc (pc , d) ≤ LTE , ∀ Contingency c

(9)

fc (p, d)
|p − pc |

≤ R15 , ∀ Contingency c

(10)

where pc is the vector of unit redispatch under contingency c, fc () is the vector of
power flows under contingency c, and R15 is the vector of units’ 15-minute ramp
capabilities. The corrective actions in the above formulation are the unit redispatch
under each contingency c. The goal of the corrective actions is to retain the
contingency power flow below LTE (9). The corrective actions are constrained by
the unit’s ramping capability (10). By considering the corrective redispatch actions
pc , the power flow immediately after the contingency is relaxed from LTE in (4) to
STE in (8), thus reducing the dispatch cost. From a mathematical perspective, the
introduction of corrective actions pc in (5)–(10) allows a larger feasibility region for
the dispatch decision p than the original SCED formulation (1)–(4). This is due to


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the fact that the corrective SCED will turn into the conventional SCED if one fixes
the redispatch variables pc to p.
Compared to the conventional SCED, the numbers of variables and constraints
of the SCED with corrective redispatch increase dramatically by a factor of N (the
number of contingencies). The solution of such a problem, in particular for realtime applications, is challenging. Decomposition techniques would have to be used
together with parallel computing. Significant progress has been made on solving
such problems [12, 13], and the latest reported results show that the problem can be
tackled within several minutes for a large power system [14].

5 Modeling Uncertainty in Grid Operation
Uncertainty caused by the renewable integration is a key element of the system
complexity. How to manage the system change caused by the sudden wind
drop, cloud covering of solar panels, and high-speed wind cutout becomes an
important field of study. Several methods exist today: deterministic method with
increased operating margins such as additional reserve and ramp requirements,
stochastic optimization, robust optimization, and chance-constrained optimization.
The deterministic method is simple, but its efficiency is heavily dependent on
the operating margin selected. Recent studies have shown that both stochastic
and robust optimization techniques can achieve better efficiency in the uncertainty
management. In this section, we first present the deterministic approach and then
discuss two techniques in the process of making unit commitment (UC) decisions
under uncertainty.

5.1 Deterministic Unit Commitment
A unit commitment problem can be stated as the system operator finding the optimal
schedules of resources over a short time period, typically 24 hours for a day-ahead
market or 1–4 hours for the real-time operation under the ISO environment, based on
a cost minimization principle. For a deterministic UC problem, the optimal solution
must satisfy the physical characteristics of resources, a set of operating constraints,
and the demand forecast. A generalized deterministic security-constrained UC
(SCUC) problem can be formulated as the following compact matrix form:
minx,y cT · x + bT · y , s.t.

(11)

Ax + By ≤ g ,

(12)

Hy ≤ h ,

(13)

Id y = d¯ ,

(14)


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Fx ≤ f ,

(15)

y ≥ 0, x is binary .

(16)

where x is the vector of binary commitment-related decision variables that may
include a unit’s on/off status and start-up or shutdown variables. c is the vector
of the commitment costs that include the start-up cost and no-load cost. y is
the dispatch decision variable that includes energy dispatch and ancillary service
dispatch from both generators and loads, and b is the vector of the incremental
energy and ancillary service costs. Equation (12) represents the coupling constraints
between the commitment decisions and dispatch decisions, e.g., units’ maximum
and minimum operating limits and start-up and shutdown ramps. A , B and g are
the coefficient matrixes and parameter vectors associated with (12). Equation (13)

represents the dispatch constraints, e.g., reserve requirements constraints, transmission constraints, units’ ramp limits, energy and reserve capacity constraints, etc. The
equality constraint (14) corresponds to the expected energy balance constraint. Id is
an indicator matrix that selects the components of vector y to meet the expected
demand d. (15) represents constraints related to the commitment decisions, e.g.,
units’ minimum up and down constraints, start-up cost constraints, etc. F and f are
the coefficient matrix and the limit vector for (15).
Deterministic UC problem is often formulated as a mixed integer linear programming problem, which can be solved efficiently by commercial MILP solvers or
Lagrangian relaxation method.

5.2 Stochastic Unit Commitment
Different from the deterministic UC, which determines the commitment schedule
to meet the expected system condition such as the expected system load and
the expected renewable generation, the stochastic optimization approach explicitly
incorporates the probability distribution of the uncertainty [15–17]. A general form
of a two-stage stochastic UC problem with the consideration of random system
demand can be represented as
minx,y cT · x + E(bT · y(ω)) , s.t.
Ax + By(ω) ≤ g ,
Hy(ω) ≤ h ,
Id y(ω) = d(ω) ,
Fx ≤ f ,
y(ω) ≥ 0, x is binary .

(17)


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Compared to the deterministic UC, the objective function of the stochastic UC
contains two parts: the first-stage commitment cost cT x and the expected secondstage dispatch cost E(bT y) . E() is the expectation function over the random event
ω. y(ω) is the recourse action or the dispatch solution in event ω. The first-stage
decision is the commitment variable x, and the second stage decision is the dispatch
solution y(ω), which has to meet the random demand realization d(ω).
Many methods exist in solving the stochastic UC problem. [18] adopted the
progressive hedging method, [19] utilized the Lagrangian decomposition technique.
The most common solution technique is the Benders decomposition, where the
master problem and subproblems are solved iteratively until convergence. The major
limitation of stochastic UC in applying to large-scale power systems is the need
for probability distribution of random variables and the possible large number of
scenarios that requires intensive computation.

5.3 Robust Unit Commitment
Robust optimization has recently gained substantial popularity as a modeling
framework for optimization under uncertainty, led by the work in [20–26]. The
approach is attractive in several aspects. First, it only requires moderate information
about the underlying uncertainty, such as the mean and the range of the uncertain
data; and the framework is flexible enough that the modeler can incorporate more
probabilistic information such as the correlation to the uncertainty model, when such
information is available. Second, the robust model constructs an optimal solution
that immunizes against all realizations of the uncertain data within a deterministic
uncertainty set. Hence, the concept of robust optimization is consistent with the
risk-averse fashion in which the power systems are operated.
Following the decision-making process (UC decision before the operating day
and the dispatch against the uncertainty realization), we extend the previous
deterministic formulation and discuss a two-stage adaptive robust unit comment
model that considers adaptive economic dispatch actions in the real-time operation
and produces robust commitment solutions to account for the uncertainty in the
individual load. In this model, demand is assumed to belong to a polyhedral

uncertainty set, which can be represented in the following general form:
D ≡ {d | M · d ≤ N, d ≥ 0} .

(18)

Therefore, we replace (14) in the deterministic model by the following equation:
yi,t = di,t , ∀(i, t) ∈ L × J where di,t is uncertain demand level and d ∈ D.
The two-stage adaptive robust UC model is formulated as follows:
minx (cT x + maxd∈D miny∈{y| By≤g−Ax, Hy≤h, Id y=d, y≥0} bT y) , s.t.
Fx ≤ f , x is binary .

(19)


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The first-stage decision variables are the binary decisions that are related to the unit
commitment. The system operator implements the unit commitment (here-and-now)
decision before the observation of the actual load values. The power outputs and
reserves are the second-stage (wait-and-see) decision variables, which are chosen
after the uncertainty is realized. The goal of the above adaptive UC model is to find
a robust unit commitment decision that minimizes the sum of the commitment costs
for first-stage decisions and the worst-case dispatch costs induced by the first-stage
together with the second-stage decisions.
Uncertainty set is an important aspect of the robust optimization. Different
characterization of uncertainty set can affect the conservativeness and thus the
solution of a robust optimization problem. Uncertainty sets described by different
norms and the concept of uncertainty budget are discussed in [27]. To reduce the

conservativeness of the robust optimization, some researchers adopt the data-driven
approach in constructing the uncertainty set, which could also incorporate the spatial
and temporal correlation of uncertain parameters.
Compared to stochastic UC, robust UC does not require probabilistic information
about the uncertainty and tries to minimize the worst dispatch cost rather than the
expected dispatch cost. The computation effort is relatively small. Methods used
in the stochastic UC can be used to solve the robust UC problem. These methods
include Benders decomposition, column and constraint generation, and affine policy
approximation of the adaptive actions.

6 Managing System Flexibility
As more variable resources are integrated into the electric power system, supply
and demand uncertainty increases dramatically. This requires the system to have the
ability to react to sudden changes and accommodate new status within acceptable
time period and cost. Therefore, the notion of flexibility recently has been drawing
extensive attention in the power industry.
Most of the flexibility definitions in the literature [28–33] and metrics proposed
pertain to particular aspects of power systems. Many of the assumptions underlying
some of the metrics make their field of application very narrow. A unified flexibility
framework for power systems is needed and will allow flexibility to be explicitly
considered in the design of the system from both short-term and long-term
perspectives and in control algorithms. In this section, we identify four elements,
response time window, uncertainty, course of action, and cost, that are common to
the flexibility literature in power systems. These four crucial elements serve as a
basis for constructing effective measures of flexibility that can be applied to a wide
range of situations.


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6.1 Definition of Flexibility
Flexibility at a particular state is the ability of the system to respond to a range of
uncertain future states by taking an alternative course of action within acceptable
cost threshold and time window. Flexibility is an inherent property of a system. The
following four elements are identified as the determinants of the flexibility: response
time window (T), set of corrective actions (A), uncertainty (U), and response cost
(C). The first three elements are affected by the power system operating criteria
while the last element is determined by the economic criteria. Next, we will describe
each element in detail.

6.1.1

Response Time Window (T)

The response time window indicates how fast the system is expected to react to
the state deviations and restore the system to its normal state. The time window
can be seconds, minutes, hours, days, or months depending on the purpose of the
study. Based on the selected response time, a system may have different flexibility
levels. Shorter time windows focus on the short-term operational flexibility, which
indicates a system’s timely response to emergency in minutes or hours. Longer time
windows focus on the long-term planning flexibility, which shows a system’s ability
to cope with changes such as generation mix, regulatory policy, and electricity
consumption pattern changes in years. Therefore, the time horizon has to be
determined when we compare and evaluate system flexibility.

6.1.2

Set of Corrective Actions (A)

The set of corrective actions A represents the corrective actions that can be taken
within the response time window under certain operating procedure. Therefore,
the corrective actions set depends on the response time window T, i.e., A(T). For
instance, if T=1 hr, the corrective action set may include actions such as voltage
control, commitment of units, and interchange scheduling. The size of the available
corrective action set reflects the diversity of corrective actions. The larger the set
A(T) is, the more options operators have to respond to unexpected events. In turn, the
response cost can be reduced or more uncertainty can be accommodated. Operating
procedure changes or technology improvement will affect the corrective action set.

6.1.3

Uncertainty (U)

Uncertainty is the lack of complete information of the state of the system in
the future. There has always been uncertainty in power systems operations and
planning. Uncertainty is traditionally associated with the likelihood of failure of