Power Systems as Dynamic Networks

Power Systems as Dynamic Networks David J. Hill

Guanrong Chen

Department of Information Engineering The Australian National University ACT 0200, Australia [email protected]

Department of Electronic Engineering City University of Hong Kong Hong Kong, China [email protected]

Abstract—The theory of power systems dynamics has been developed largely from detailed studies of the dynamics of simple system structures with emphasis on the effects of modeling details of the dynamics. In contrast, the recent study in the science of complex networks, motivated by numerous real-world examples in various areas including power systems, employs rather simple dynamics and yet places much more emphasis on network structures. It appears that the two approaches can usefully inform each other for further progress into an integrated study of very large (or massive) systems as well as very complex dynamics. This paper reviews recent progress in these areas and suggests fruitful lines of future research. I.

emphasis on stability analysis, and recent results in network science. The theory of dynamics for power systems has been developed to an advanced state following contributions dating back to the 1930’s. There are theories for the major phenomena using various mathematical techniques. Of particular importance are synchronization, oscillations, bifurcations, collapse and chaos [1]. The theories have emerged largely from detailed studies of dynamics of simple system structures, often just with one or two generators, with emphasis on the effects of modeling details of the dynamics. In recent years, there has been an explosion of scientific effort aimed at study of complex networks with applications in numerous areas across physics, computer and life sciences, and engineering systems such as power systems. A major topic is synchronization phenomena. This work employs rather simplistic dynamics and places much more emphasis on the network structure including those becoming popular recently in the social and physical sciences, i.e. small-world and/or scale-free models. It appears that the two approaches, which will be referred to respectively as dynamics oriented and structure oriented, have many connections.

INTRODUCTION

Power systems are naturally modeled as large-scale and highly nonlinear structured dynamical systems. These nonlinear models can be shown to exhibit a wide variety of complex behaviour described mathematically in terms of Lyapunov stability, structural stability, bifurcations, and chaos. Previous planning and operating practice has been able to proceed largely without deep concern for these aspects of complex system behaviour. Reference [1] in 1995 gave a presentation of the state-of-art in nonlinear phenomena and predicted that in the future they will be more important to power systems. Economic and environmental restrictions, along with the trend to open access, have caused operating margins to be reduced as systems become more heavily loaded. Consequently, system behaviour will be more dependent on nonlinear characteristics. Recent experience, especially a number of major blackouts worldwide in 2003, suggests that we have entered an era of less predictable behaviour in more complex power networks. We now talk of power systems as complex networks with a tendency to catastrophic failures [2]. This in turn requires much more sophisticated analysis. Generally, we can be confident that the scientific basis for operating large systems like power systems needs to be strengthened. This paper explores the possible synergy between the nonlinear theory of power systems, with

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II.

POWER SYSTEMS STABILITY THEORY

From a practical viewpoint, there are four major analytical problems: a) using power flows to compute equilibria; b) applying angle stability (commonly called transient stability or large-disturbance stability) to check preservation of generator synchronism; c) studying oscillations (commonly treated as a linear issue); and d) employing voltage instability to check the possibility of voltage collapse. Of course, theoretically they are all aspects of the one overall stability question [3], but it is possible to focus on each via appropriate modelling simplifications. A power system consists of generators (power sources) and loads (power sinks) connected to buses which are interconnected by transmission lines in a network structure. For transient stability, i.e. the electro-dynamical behaviour, the loads are usually taken as static. Most papers present this model with the loads taken as impedances and the network

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modelled in reduced forms, i.e. the load buses absorbed into the network. Following the work by Bergen and Hill [4, 5] and subsequently (see for instance [6-9]) we use a networkpreserving model (NPM) here. The generators in the system are modelled as constant voltages behind a reactance. The phase of this voltage is determined by the rotor dynamics with respect to a synchronous reference and is modelled by the swing equation (1) M i θ&&i + D i θ&i + VV i j bij sin(θi − θ j ) = P i

commonly embedded in a stability question for the postfault equilibrium point, i.e. the post-fault angle differences and frequency must lie in a region of attraction of that equilibrium; c) the tendency to get undamped oscillations in frequencies ω i , which are precisely analysed as Hopf bifurcation, must be avoided; d) the voltages Vi are subject to different problems on different time-scales including short-term [8] and long-term voltage collapse [3], which are analyzed in terms of local and global bifurcations (see [1] and references therein). The earliest contributions to power systems theory recognized the highly nonlinear nature of (3). We now list some points of the dynamical systems theory on power systems which relate to the theme of this paper:

∑ j∈Ci

This expresses a dynamic real power balance. The subscript i denotes machine i and j ∈ Ci refers to the machines connected to machine i . At each network bus, the voltage magnitude and phase are variables. These buses generally have loads attached. Adding equations for the associated power balance gives, for each load k ,

∑V V b k

j kj

•

sin(θ k − θ j ) = Pk (Vk ) (2)

•

The key symbols in (1)-(2) are as follows: θ is the bus voltage phase (including internal voltages of generators); V is a bus or nodal voltage (rms value); bij is the admittance of the line joining nodes i and j ; M is the rotor’s inertia constant; D is the damping coefficient; P and Q are the real and reactive power input to a bus (which for the generator is the mechanical power applied to the rotor shaft). For the loads, the powers are shown as voltage dependent to reflect the effect of load characteristics). There are two state variables associated with each generator, i.e. angle θi and

•

j

−∑ VkV j bkj cos(θ k − θ j ) = Qk (Vk ) j

•

frequency ω = θ& . If loads are modeled as dynamic, there i i will be other dynamic state variables corresponding to internal behavior of the loads, e.g. motor slips. These equations are typically assembled into a higherdimensional set of differential algebraic equations (DAE) of the form

x& = f ( x , y ) 0 = g ( x, y )

•

•

(3)

The (dynamic) state x includes generator angles, frequencies and maybe higher-order effects in the generator dynamics and load dynamics. The algebraic variables y include load bus voltages. The above-mentioned stability questions can be summarised as follows (see [3] for more detail): a) solution of (3) in steady state, modulo some subtleties [10], provides the equilibria which of course depend on the power flow imposed on the system; b) the generator angle differences θi - θ R , where θ R is a reference defined by one

There are rigorous Lyapunov functions for NPMs where the real-power loads are independent of voltages [4-6, 11]; this had proved problematic for decades with reduced models; There are extensive theory and algorithms for characterizing and computing the regions of stability around stable equilibria; The Lyapunov (or energy) functions can be expressed in a topological form [4], i.e. a sum of contributions from all generators and lines; denoting the line angle differences by σ k = θ i − θ j , the Lyapunov function can be written 1 (4) E(ω,V ) = ∑ M iωi2 + ∑Wk (σ k − σ k0 ) G 2 L The vulnerability of the network to large disturbances in certain loading conditions can be related to network cutsets [4, 12] (although a complete theory has not been developed); Allowing for multiple equilibrium points and the local solvability of (3), and depending on the load characteristics, the energy function can be seen as defined over multiple hypersurfaces corresponding to different stable equilibria and voltage solutions [8]; In some cases, this application inspired extensions to the basic stability theory, e.g. [11].

III.

NETWORK SCIENCE

Most systems in nature and engineering can be described by a network consisting of nodes connected by links, e.g. the buses and lines respectively in a power system reviewed above. Engineers tend to describe complexity in terms like dimension, nonlinearity, uncertainty and structure [13]. Scientists are more likely to mention structural features, more specifically e.g. diversity, evolution, self-similarity or behavioural features like emergence even though many of these concepts remain somewhat difficult to define [14]. Here, we focus on results related to dynamic networks

machine or some specially defined angle, e.g. centre of inertia, must remain small to maintain synchronous operation in the presence of large disturbances; this is

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described by differential equations and results on network synchronisation (see [15, 16] for surveys). From [16], we obtain a typical model and some stability results. Consider a dynamical network consisting of N identical and linearly coupled nodes, with each node described by an n-dimensional dynamical system. The dynamics are described by N

x& i = f ( x i ) + c ∑ aij Γx j , i = 1,2,3..., N

Network science has identified classes of structures which are commonly observed in the real world. These include small-world and scale-free networks [14-16]. The latter property has been shown to apply to numerous largesize networks including the Internet, WWW, metabolic networks and power systems. It refers to absence of scale in the sense of connectivity of the nodes, i.e. a few nodes have high connectivity (or hubs) and numerous nodes have low connectivity. The current synchronization literature has, and still is producing large numbers of results in the spirit of (6) on synchronization of such networks with structure (5); the generalizations are too numerous to list here, but include time-varying, nonlinear, and delayed interconnections (see e.g. [19, 20]). For example, these results infer the ability to enhance synchronization in a large-size nearest-neighbor coupled network by just adding a tiny fraction of distant links, thereby making the network become a small-world model. The robustness of synchronization in a scale-free dynamical network has also been investigated, against either random or specific removal of a small fraction of nodes in the network. Although the scale-free structure is particularly well-suited to tolerate random errors, it is also particularly vulnerable to deliberate attacks. The error tolerance and attack vulnerability in scale-free networks are rooted in their extremely inhomogeneous connectivity patterns. There are further topics in network science that relate to power systems issues. In particular, we mention the need to understand cascading collapses [21] and how to design networks to be optimal in desired senses, e.g. for synchronizability [22, 23].

(5)

j =1

where xi = ( xi1 , xi 2 ,...xin ) is the state at node i, constant c>0 represents the coupling strength and Γ is an internal constant connectivity matrix for the variables of each node. The constant coupling matrix A = (aij) represents the coupling configuration of the network. If there is a link between node i and node j (i≠j), then aij = aji = 1; otherwise, aij = aji = 0 (i≠j). Clearly, aii = −ki, where ki is the degree of node i, defined as its weight or commonly as the number of links it has. Network (5) is said to be (asymptotically) synchronized if x1(t) → x2(t) → · · · → xN(t) → s(t), as t→ ∞ where s(t) is a solution of an isolated node, i.e. s&(t ) = f ( s (t )) . Here, s(t) can be an equilibrium point, a periodic orbit, or a chaotic orbit, depending on the nature of the study. Consider the case where the network is connected in the sense that there are no isolated clusters. Then, the coupling matrix A is symmetric and irreducible. In this case, its eigenvalues satisfy 0 = λ1 > λ2 ≥ λ3 ≥ ··· ≥ λN. It has been shown that the synchronization state is exponentially stable if c ≥ │ d / λ2│ (6) where d < 0 is a constant determined by the dynamics of an isolated node and the internal node structural matrix Γ. The result (6) is obtained as a byproduct of synchronization theory for coupled systems (motivated by arrays of chaotic oscillators) [17]. Under a different setting, this condition may not be the same [18]. In contrast to earlier stability theory of interconnected systems, the emphasis here is on structured and possibly strong coupling. The results, for local (around a trajectory in the state space) stability, infer that the synchronizability of the dynamical network (5) can be characterized by the second-largest eigenvalue of its coupling matrix, i.e. λ2, or by the ratio of λN/λ2 under slightly different conditions [18]. The beauty of this result is that the eigenvalue(s) can be neatly related to network structural features. For instance, λ2 for a globally coupled network is −N, which implies that for any given and fixed nonzero coupling strength c, a globally coupled network will synchronize as long as its size N is large enough. On the other hand, the λ2 of a nearest-neighbor coupled network tends to zero as N →∞, from which one may infer that for any given nonzero coupling strength c, a nearest-neighbor coupled network cannot synchronize if its size N is sufficiently large. These simple results are interesting and also significant in that they point to a theoretical possibility of cleanly relating stability properties to system structure in more general dynamical networks.

IV.

FUTURE RESEARCH OUTLOOK

The two theories that have been described in Sections II and III respectively have considerable commonality in the problem studied. They are both essentially about synchronization in networked systems. Generally speaking, the power systems results are more oriented to node dynamics while the complex networks results are oriented to system structures. However, neither theory can be currently used to make statements about the other. It appears that the theories provide two approaches, which will be referred to as dynamics oriented and structure oriented, respectively, and they can very usefully inform each other for further progress into the study of very large (or massive) systems and very complex dynamics. Some issues which could be further studied are: • The power systems theory needs to be pushed further in the direction of exploiting structural features of the networks; for instance how to exploit a scale-free structure (or a refined version) in dynamical analysis; there are challenges in this task; realistic power systems models have at least two different types of node dynamics (generators, loads) and the directional power flows between them play a major role;

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[4] A.R.Bergen and D.J.Hill, “A structure preserving model for power system stability analysis”, IEEE Trans Power Apparatus and Systems, Vol. PAS-100, No. 1, pp. 25-35, January 1981. [5] D.J.Hill and A.R.Bergen, “Stability analysis of multimachine power networks with linear frequency dependent loads, IEEE Trans on Circuits and Systems, Vol. CAS-29, No. 12, pp.840848, December 1982. [6] N.Narasimhamurthi, “A generalized energy function for transient stability analysis of power systems,” IEEE Trans on Circuits and Systems, Vol.CAS-31, No.7, pp.637-645, July 1984. [7] M.A.Pai, Energy Function Analysis for Power System Stability, Boston: Kluwer AcademicPublishers, 1989 [8] I.A.Hiskens and D.J.Hill, “Energy functions, transient stability and voltage behaviour in power systems with nonlinear loads”, IEEE Trans on Power Systems, Vol. 4, No. 4, pp. 1525-1533, November 1989. [9] H-D.Chiang, “Direct stability analysis of electric power systems using energy functions: theory, applications, and perspective”, in [1]. [10] D.J.Hill, “On the equilibria of power systems with nonlinear loads”, IEEE Trans Circuits and Systems, Vol. CAS-36, No. 11, pp. 1458-1463, November 1989. [11] D.J.Hill and I.M.Y.Mareels, “Stability theory for differential/ algebraic systems with application to power systems”, IEEE Trans Circuits and Systems, Vol. CAS-37, No. 11, pp. 14161423, November 1990. [12] K.S.Chandrashekhar and D.J.Hill, “A cutset stability criterion for power systems using a structure preserving model”, Intl Journal of Electrical Power and Energy Systems, Vol. 8, No. 3, pp. 146-157, July 1986. [13] D.D.Siljak, “Complex dynamic systems: dimensionality, structure, and uncertainty,” Large Scale Systems, Vol.4, pp.279-294, 1983. [14] A.L.Barabasi, Linked, Plume, 2003 [15] S.H.Strogatz, “Exploring complex networks,” Nature, Vol. 410, pp.268-276, 8 March 2001. [16] X.F.Wang and G.Chen, “Complex networks: small- world, scale-free and beyond, ” IEEE Circuits and Systems Magazine, First Quarter, pp.6-20, 2003. [17] C.W.Wu and L.O.Chua, “Synchronization in an array of linearly coupled dynamical systems,” IEEE Trans Circuits and Systems-I, Vol.42, No.8, pp.430-447, August 1995. [18] M.Barahona and L.M.Pecora, “Synchronization in smallworld systems”, Phys. Rev. Lett., Vol.89, No.5, pp.054101-4, 2002. [19] J.Lü and G.Chen, “A time-varying complex dynamical network model and its controlled synchronization criterion,” IEEE Trans. on Auto. Contr., Vol.50, pp. 841-846, June 2005 [20] Z.Li and G.Chen, “Synchronization in general complex dynamical networks with coupling delays,” Physica A, Vol. 343, pp.263-278, Nov. 2004 [21] J.Xu and X.F.Wang, “Cascading failures in scale-free coupled map lattices,” Physica A 349, pp.685-692, 2005. [22] J.Lü, X.Yu, G.Chen and D.Cheng, “Characterizing the synchronizability of small-world dynamical networks,” IEEE Trans Circuits and Systems-I, Vol.51, No.4, pp.787-796, April 2004. [23] M.E.J.Newman, M.Girvan and J.D.Farmer, “Optimal design, robustness and risk aversion,” Physical review Letters, Vol.89, No.2, July 2002. [24] D.J.Hill and P.J.Moylan, “General instability results for interconnected systems”, SIAM Journal of Control and Optimization, Vol. 21, No. 2, pp. 256-279, March 1983.

• The power systems theory needs to follow the ideas of collapse dynamics in networked systems; • The networks results are typically for the same dynamics at each node; they need to be extended to classes of node dynamics, e.g. same form but different parameters as in power systems theory; • Further possibilities are in the use of properties like passivity and small-gain on the node dynamics, bringing a connection to results on the stability (and instability) of large-scale systems [24]; • The networks results are typically local, i.e. around a trajectory in the state space; power systems theory has characterized regions of stability (and synchronization) and even global behavior; the tools of absolute stability theory, normal forms, bifurcations and chaos have all been used; • In the application of Lyapunov methods, there would appear to be value in using different forms such as Lur’e-Postnikov, weighted sums, and so on; • The networks results are focused on synchronization; as the power systems case illustrates, there are other dynamical phenomena that are related to structures, i.e. collapse phenomena, oscillations, and collapse; • Many networks such as power systems and the Internet involve some underlying flows which determine equilibria and bifurcations; this needs to be brought into the models used for generic results; • There are some steps towards robustness analysis in networks, but more refined studies are needed, i.e. the effect of link switching on regions of convergence • In both areas, attention needs to be given to the hybrid nature arising from switching events and how this integrates with the dynamics and the network. V.

CONCLUSIONS

This paper has reviewed the theories of power system stability and dynamical network synchronization. While sharing considerable overlap in their concern for synchronous behavior, the results differ in emphasis. There appear to be some very fruitful synergies to pursue for progress in power systems, and networks generally, especially towards large complex systems.

REFERENCES [1] D.J.Hill (ed), Special Issue on Nonlinear Phenomena in Power Systems: Theory and Practical Implications, IEEE Proceedings, Vol. 83, No. 11, pp. 1439-1587, November 1995. [2] P.Fairley, “The unruly power grid,” IEEE Spectrum, August 2004, pp.16-21. [3] P.Kundur et al., “Definition and classification of power system stability” , IEEE Trans Power Systems, Vol.19, No.3, August 2004, pp.1387-1401.

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