Abstract— Vision of a fail proof smart grid can be fulfilled using modern control capabilities. To maximize performance on the “edge”, systems often evolve into a subtle mix of order and randomness (e.g., biology or astrophysics). Systems exploiting such “deterministic unpredictability” have been built (e.g., high performance aircraft). This paper proposes to expand the envelope of stable operations into the realm of “managed chaos” in order to realize an agile power grid that is more efficient and reliable with less redundancy in transmission capacity. This involves operation of the system with a string of local ephemeral equilibria created by changing control objectives, strategies and/or algorithms. The shallow potential wells around these equilibria keep the system prone to managed chaos, until reaching an acceptable final equilibrium. Overcoming grid stability problems as proposed would enable end-to-end direct transactions between energy producers and consumers with internet-like market reach, transparency and ease of execution.

Index Terms— flexible smart grid, power and information market, relaxed stability, power system stability, power system reliability and control, fast local control, distributed autonomous intelligence, coordinated local and wide area control, ephemeral equilibrium.

I. INTRODUCTION ISION of a fail proof power grid remained elusive

since the inception of the grid. Most large-scale disturbances unfold as a result of unfortunate sequences of adverse events involving acts of nature, accidents and human errors. In addition, as power systems are getting interconnected with information system devices, applications and networks in order to enhance efficiency, increase state transparency and bring about distributed and adaptive control, their vulnerability to deliberate attacks also increase significantly. Almost always, such sequences have a high likelihood of culminating in a precipitous spontaneous cascade of system configuration changes driven by loss of stability in each fleeting configuration. Hence, overcoming stability problems is essential for realizing the vision of a fail proof grid.

Current practices in dealing with stability problems involve

a priori simulation and analysis of all postulated credible adverse events and identifying boundaries of safe operation in the current operating configuration to withstand any postulated event if and when it actually happens. This preemptive approach results in a conservative operating posture with consequent under utilization of resources. However, the ability The authors are with Albeado Inc, Saratoga, CA. They can be reached at ranjit.kuma@albeado.com, christopher.reed@albeado.com and partha.dattaray@albeado.com

to successfully mitigate such stability problems after the actual occurrence of a disturbance event would allow an expanded range of pre-disturbance operating conditions allowing for a more optimal operation.

Modern measurement devices (e.g., synchrophasors)

coupled with FACTS (Flexible AC Transmission System) devices allow unprecedented control capabilities that can be exploited to overcome stability problems as and when they happen. This paper addresses the implications of this capability to the operating environment in the smart grid.

The remainder of the paper is organized as follows. Section

II provides a glimpse of the historical drive towards unfettered long distance transmission of power. Section III presents an overview of chaos in power systems. Section IV discusses the motivation to expand the operating domain into “managed chaos”. Section V presents a qualitative description of a representative control algorithm that can help in stabilizing the system in that domain. Section VI discusses the relationship between conventional stability and managed chaos. Section VII and VIII contain conclusion and references respectively.

II. HISTORICAL BACKGROUND There have always been power technologists and

enthusiasts dreaming of unfettered transmission of power over long distances. In the arena of traditional bulk power transmission, a vision of cross continental power transmission system was proposed by Baum in 1921 [1] based on a 220kV transmission system with almost constant voltage maintained by enough reactive resources placed every 100 miles (160 km). This was intended to make “power transmission comparable to railway transportation, with a flexibility not possible in the ordinary system which does not have the constant voltage feature” [1]. This vision was revisited by Kimbark in 1983 [2] with the optimistic note: “Now, however, by rapid control of voltage, we can greatly improve the transient behavior”. With modern technologies providing sub-second measurements and controls, now it is possible to contemplate a smart grid with the above envisioned “flexibility” that is physically and economically feasible.

Stability problems are categorized by the nature of their

initiating disturbances into voltage, transient and dynamic (small signal) stability. In [3] a “Baum-Kimbark” stability criterion was defined to assure voltage stability in the smart grid. In [4] a method to assure transient stability in a smart grid is presented. In this paper we discuss the operating implications of assured transient stability for system

Induced Chaos for an Agile Smart Grid Ranjit Kumar, Senior Member, Christopher Reed Member, Partha Datta Ray, Member

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operations in the smart grid.

III. CHAOS IN POWER SYSTEMS Formally, chaotic behavior is characterized by topological

mixing and dense periodic orbits [5]. Informally, this implies that small changes in the initial conditions lead to a large changes in the outcomes rendering even deterministic systems unpredictable in the long term [6]. Any system with at least three or more state variables and at least one non-linear interaction has the potential to display chaotic behavior. In this context, a state variable can be an energy storage mode in a physical element (e.g., kinetic or potential energy of a generator associated with the two state variables of phase angle and speed) or a delayed response in case of a computational element (e.g., analog circuits or computing algorithms). A large scale power system has a large number of state variables as well as a large number of non-linear interactions and hence obviously has potential for chaotic behavior.

The possible existence and onset of chaos in power systems

has been investigated (e.g., see [7, 8, 9]). However, most consider chaos as something to be avoided in the context of power systems. This is because the traditionally available control mechanisms were too slow to deal with transient oscillations, let alone chaos.

When systems are pushed to their performance limits,

chaos becomes a fact of life and in general the required speed of response generally exceeds human capabilities. Automated fast control responses are necessary to exploit the associated capabilities of the system. Aircraft design has evolved into this domain (e.g., [10]). The crux of the feasibility of managed chaos in systems is the presence of reliable automated response capability. The evolution of smart grid with its dependable IT infrastructures presents an opportunity to explore the exploitation of managed chaos.

Chaotic systems usually have multiple stable equilibria with shallow potential wells where the typical energy of the system is greater than the amount required to climb across the unstable equilibria between the stable equilibria. The shifting of trajectories from the influence of one stable equilibrium to another appears as chaotic behavior. Chaotic behavior in a typical power system is rare because it usually has a stable equilibrium point that is vastly more preferable than other stable equilibria. In addition, the preferred equilibrium point has a deep potential well and requires a considerable amount of energy to move out of the potential well and into another equilibrium position. In a rhetorical sense, even an unstable system would eventually reach another stable equilibrium, albeit an unacceptable one.

As the load on a power system increases, the corresponding

stable equilibrium becomes shallower and the kinetic and potential energies associated with disturbances increases. This leads to increased probability of instability. This is illustrated

in Fig. 1 below. The figure presents the potential energy of a single generator connected to an infinite busas a function of the instantaneous phase angle. The three curves in the figure correspond to three different post-fault steady state loading levels with the corresponding stable equilibrium points (δ) at 0, 30 and 60 degrees respectively.

Fig.1: Equilibrium and Potential Well

IV. NEED FOR MANAGED CHAOS IN POWER SYSTEMS In [11] a vision of the smart grid from the market

perspective was presented. In that vision, stability problems would be completely subdued to make the power grid as flexible as any other commodity network such as natural gas, water, railways, telecommunications and the internet as well as stock exchange. This will enable end-to-end direct transactions between energy producers and consumers with internet-like market reach, transparency and ease of execution. The vision was based on several premises including the following:

Premise (a): Ultimately, energy storage capability of all

scales will be pervasive in the smart grid. Premise (b): Ultimately, the command and control

techniques will be self-similar at all scales in the smart grid. Premise (c): Ultimately, sub-second control responses will

be deployed pervasively in the smart grid. Premise (d): Ultimately, in the smart grid almost all

system components can be operated to their thermal limits at any time.

Premise (e): Ultimately, in the smart grid, the function of

the centralized control centers would be limited to enforcement of “rules of road” rather than real-time control of the system.

In typical transmission lines, stability limits, thermal limits and power losses decrease in inverse proportion to length, conductor radius and conductor cross section area respectively. Since these three physical dimensions of the line

are not proportional to each other, sometimes the stability limits fall below the thermal limits leading to stability constrained systems. Distributed decision intelligence orchestrating properly coordinated sub-second control responses, supported by ubiquitous storage resources, allows power system operation with stability limit(s) greater than the corresponding thermal limit(s) at all times [1, 2, 3 and 4]. Such complete avoidance of the stability problems through autonomous automated controls will allow operating strategies to maximize transmission asset utilization and/or minimize transmission losses, etc.

However, to minimize costs, the modified stability limits

will be only minimally above the thermal limits and loading levels would be very close to the thermal limits for several resources simultaneously. This leads to very shallow equilibria which would be unacceptable under traditional operating procedures. In [4], it is suggested that a power system can be operated at a sequence of ephemeral equilibria until an acceptable steady-state operating point is reached. Ephemeral equilibrium is defined as an unstable equilibrium point of the open loop system where stability can be maintained if and only if appropriate closed loop controls exist. Thus essentially, the system can be operated in a region in which the open loop system is unstable. This condition is also known as “relaxed stability” in the aircraft industry.

V. STABILITY CONTROL FOR THE SMART GRID

The flow along a particular path (a sequence of branches) in the grid can be increased by supplying additional reactive power at selected points along the path. Conversely, the flow can be decreased by withdrawing reactive power supply at selected points. Thus installing adequate dynamically controllable reactive sources at selected points would allow considerable control over the power flows in the various paths of the grid. Modern control capabilities involving synchrophasor measurements and SVC’s can provide fast control signals (>20Hz) necessary to implement the required control capability. Performance of such power flow control mechanisms are described in [3] and [4]. Using such controls one can load any transmission line to any desired level (e.g. thermal limit, or a lower level that minimizes system wide losses) without any concern about voltage stability or transient stability.

In order to realize such operational flexibility, the

following two control requirements should be satisfied:

(a) Steady-state control requirements: There should be adequate dynamically controllable reactive resources (e.g. SVCs) to maintain a system-wide scheduled voltage (e.g., 1 p.u.) at all “pilot” buses. In the steady-state, the phase angles between the pilot buses should be less than a theoretical maximum of 60 degrees.

Appropriate dead bands and safety margins can be applied to provide adequate margins for measurement errors, time delays, etc. Whenever voltage at a pilot bus falls below 1 p.u. the reactive power injection at that bus needs to be increased and conversely, whenever the voltage exceeds the required value, reactive power injection should be decreased. The response time for such control signals should be very small (in the range of 10 to 50 milli-seconds) though a time skew (delay) of about 100 milli-seconds can be tolerable. This control is based on purely local bus voltage measurement only, except perhaps to prevent it from reacting to low voltages due to faults. This requirement applies to both pre-disturbance and post-disturbance steady states. This will eliminate the problem of voltage stability in the grid. See [3] for further details.

(b) Transient-state requirements:

At the mid-points between the pilot buses, there should be adequate dynamically controllable reactive power resources to handle transient swings. In the transient state, the control objective is not to maintain 1p.u voltage at that point, but to increase or decrease the voltage to a level that would help to damp the oscillations. Then the control algorithm is to increase the reactive power injection (at the mid-point) if a weighted sum of speed deviation (and calculated) acceleration is positive and decrease the reactive power injection if the weighted sum is negative. In addition, this control signal should be augmented by a small amount of reactive power injection that is in proportion to the injection in the previous time step in order to drive the reactive power injection to zero in the post transient steady-state. Note that this control algorithm depends only on a local measurement of the rate of change of the phase angle based on the time interval between the current zeros identified by a local PMU. In addition, the local measurement of instantaneous power transfer along the transmission line is used in the calculation of the feedback gains. For further details about this control algorithm, see [4]. The mid-point location is preferred because that is where a single additional reactive power source would be most economical. If there are more than one additional resource, they would be most effective if they are placed at equal intervals. No matter how many additional resources are provided their total reactive capacity would be approximately the same and depends upon the severity of the post-disturbance operating condition to be handled.

All other available control mechanisms and energy storage

devices can be used as usual. They will only positively complement the performance of the above proposed control algorithms.

VI. MANAGING INDUCED CHAOS Ignoring the conventional stability boundaries in principle

leads the system into operating conditions previously

considered unstable. This does not mean that the system is violating laws of physics. It just means that the system is stable conditional upon the closed-loop operation of the new control algorithms similar to the one proposed above.

Such conditional stability has already been in practice in

both power systems and aircraft operation throughout their history. For example, an airplane can be flying if and only if its engines are working. Power systems can function if and only if all machines connected to it are in synchronism. However we take this conditional stability for granted because they represent operation close to a single preferred stable equilibrium intrinsic to the current hardware configuration of the system. This is not chaos.

However, the proposed control mechanism can create one

of many stable equilibria in an operating domain where none could exist without the mechanism. These ephemeral equilibria can be brought into existence and made to disappear by merely changing control objectives, strategies and/or algorithms without actually changing the physical hardware. These ephemeral equilibria are relatively shallow because the system should be able to create a desired equilibrium with a minimum of energy expenditure and move into or out of that equilibrium with minimal effort. This ability to switch from the influence of one ephemeral equilibrium to another gives rise to the potential for chaotic behavior and simultaneously the agility to steer the system towards an acceptable stable equilibrium. There is no need for alarm because the ephemeral equilibria are created if and only if stability with respect to the preferred intrinsic equilibrium is threatened.

VII. CONCLUSIONS A Vision of a fail proof smart grid can be fulfilled through

the unprecedented control capabilities that are on the horizon. Expansion ofthe envelope of stable operations into the realm of managed chaos is proposed in order to realize an agile power grid that is more efficient and more reliable with less redundancy in transmission capacity. This involves the potential for operation of the system with several local ephemeral equilibrium points created by control objectives, strategies and algorithms. Since the potential wells around these equilibria will necessarily be shallow relative to the conventional stable operating points, the power system will always be prone to chaos. With appropriate control actions the power system can be steered through a sequence of ephemeral equilibria until a preferred stable equilibrium is reached.

With numerous such mechanisms in place and with

economic incentives pushing systems ever closer to their relaxed stability limits we expect to see more frequent displays of induced chaos in power systems and the challenges of managing such chaos.

Overcoming the grid stability and reliability problems as

proposed would enable end-to-end direct transactions between energy producers and consumers with internet-like market

reach, transparency and ease of execution.

VIII. REFERENCES [1] Frank G. Baum, "Voltage Regulation and Insulation for Large-Power

Long-Distance Transmission Systems," A.I.E.E Trans., Vol. 40, pp. 1018-1077, June 22, 1921.

[2] E. W. Kimbark, “A New Look at Shunt Compensation”, IEEE Trans. On PAS, vol.PAS-102, No.1, pp212-218, January 1983.

[3] Ranjit Kumar, “Assuring Voltage Stability in the Smart Grid”, PES Innovative Smart Grid Technologies Conference, January 17-19, 2011, Anaheim, CA, USA.

[4] Ranjit Kumar, “Assuring Transient Stability in the Smart Grid”, submitted to PES Innovative Smart Grid Technologies Conference, January 17-19, 2012, Washington, DC, USA.

[5] Hasselblatt, Boris; Anatole Katok (2003). A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge University Press. ISBN 0521587506.

[6] Stephen H. Kellert, In the Wake of Chaos: Unpredictable Order in Dynamical Systems, University of Chicago Press, 1993, p 32, ISBN 0-226-42976-8.

[7] I. Dobson, B.A. Carreras, V.E. Lynch, D.E. Newman “Complex systems analysis of series of blackouts: cascading failure, critical points, and self-organization”, Chaos, vol. 17, 026103, June 2007

[8] J. Chen, J.S. Thorp, I. Dobson “Cascading dynamics and mitigation assessment in power system disturbances via a hidden failure model”, International Journal of Electrical Power and Energy Systems, vol 27, no 4, May 2005, pp 318-326

[9] H. Liao, J. Apt, and S. Talukdar, “Phase transitions in the probability of cascading failures,” Electricity Transmission in Deregulated Markets: Conference at Carnegie Mellon University, Pittsburgh, PA, December 2004.

[10] H. G. Kwatny, W. H. Bennett and J. Berg, "Regulation of Relaxed Static Stability Aircraft," IEEE Trans. Auto. Contr., Vol. 36, No. 11, November 1991, pp. 1315-1323.

[11] Ranjit Kumar, Partha Datta Ray, Christopher Reed “Smart Grid: An Electricity Market Perspective”, PES Innovative Smart Grid Technologies Conference, January 17-19, 2011, Anaheim, CA, USA.

IX. BIOGRAPHIES Ranjit Kumar received Ph.D. from the University of Missouri at Rolla

(now known as Missouri University of Science and Technology). He has over 30 years of experience in research and development of algorithms and software for the design, operation and real-time control of power systems, markets and smart grid. He has made several contributions related to power system stability, fuel resource scheduling, and dynamic security analysis. He is Chief Power Application Architect at Albeado Inc.

Christopher Reed is Head of IP enabled services at Albeado Inc. and has

served in leadership roles in service and engineering organizations at companies ranging from startups to Fortune 500 including Nokia, Borland and Intellisync. Over the last 22 years, he has built and lead teams distributed around the globe to provide educational, support, integration and development services. He is presently leading the Information Modeling Subgroup for IEEE P2030 standards committee. Chris earned his BSEE at San Jose State University and is currently pursuing his JD degree.

Partha Datta Ray (M ’89) is Chief Technology Officer of Albeado Inc.

He has served in multiple technology executive roles in the past including Vice President of Engineering at GDA Technologies, Terablaze (acquired by Agere/LSI), LSI Logic, Silicon Compiler/Mentor Graphics and at AT&T Bell Labs (communications research). He holds a BSEE and MSCS from Rutgers and has conducted advanced research works on complex systems security and automation at leading institutes. He currently leads the Information Security Working Group of IEEE P2030 Smart Grid Standards Committee and has led other network and semiconductor related trade and standards groups in the past.