Stability of a distributed generation network using the Kuramoto models Vincenzo Fioriti, Silvia Ruzzante, Elisa Castorini, Elena Marchei, Vittorio Rosato
Abstract We derive a Kuramoto-like equation from the Cardell-Ilic distributed electrical generation network and use the resulting model to simulate the phase stability and the synchronization of a small electrical grid. In our model nodes are arranged in a regular lattice; the strength of their couplings are randomly chosen and allowed to vary as square waves. Although the system undergoes several synchronization losses, nevertheless it is able to quickly resynchronize. Moreover, we show that the synchronization rising-time follows a power-law.
1 Introduction One of the most important complex Critical Infrastructure (CI), the electric power system, is evolving from a ”concentrated generation” model towards a ”distributed generation” (DG) model, where a large number of small power generators are integrated into the transmission (and/or in the distribution) power supply system according to their availability. Large power plants (nuclear, coal, gas etc.) will be joined by low- (or intermediate-size) power generators, based on alternative sources (wind, solar, micro-hydro, biomass, geothermal, tidal, etc). Whereas the concentrated generation model can be (in principle) more simply controlled and managed, the DG model, with geographically unevenly distributed generation plants, producing electrical power as a function of the season, of the time of the day and the meteorological conditions, does indeed introduce, in an already complex scenario, further instability issues which are worth to be considered. More importantly, renewable source genVincenzo Fioriti, Elisa Castorini, Vittorio Rosato ENEA, Casaccia Research Center, Via Anguillarese 301, 00123 S.Maria di Galeria (Rome) Italy, e-mail: vincenzo.fioriti, elisa.castorini,
[email protected] Silvia Ruzzante, Elena Marchei ENEA, Portici Research Center, Via del Macello Vecchio, 00122 Portici (Italy) e-mail: silvia.ruzzante,
[email protected]
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Vincenzo Fioriti, Silvia Ruzzante, Elisa Castorini, Elena Marchei, Vittorio Rosato
erators insert in the network different amounts of electrical power, amounts which can be, in turn, smaller (or even much smaller) than those provided by “conventional” means (fossil sources). Developing a successful grid supporting technologies for DG requires mathematical models of interconnections, control strategies able to cope with transient effects and to produce an efficient and robust distribution system. Unfortunately, the connection of a large set of small- and intermediate-size generators to the generation and distribution network raises some problems as harmonic distortion of voltage [6], [11], [12], [13], stability of synchronization with the network, thermal limits, network faults. Moreover, in the future, other technological networks will tightly interact with the DG grid producing a tangled set of interdependencies whose final effects will undermine the stability of synchronization. Here we will focus on the development of a mathematical model, based on the Kuramoto model (KM) [2] equation, for the study of the stability of a DG grid. KM is the most successful attempt to study the ubiquitous phenomenon of synchronization, starting with Huyghens up to Van der Pol, Andronov, Wiener, Winfree, Kuramoto, Watanabe, Strogatz [3]. From simple mechanical devices to complex systems, a rich variety of dynamical systems have been modelled [8]: crowds of people, flocks of birds, school of fish, associative memories, array of lasers, charge density waves, nonlinear control systems, Josephson junctions, plasma, cardiac cells, power grids, epidemic spreading, social and economic behaviours. To derive a KM model for describing a distributed generators network, we have used the Cardell-Ilic linearized dynamic model for DG, that uses the power flow connecting the networks nodes as coupling parameter. On such a model system, we will attempt to study the effects of perturbations on the synchronization of a number of power generators (modelled as oscillators with given frequencies, different phase angles and connected with known couplings). Our model has been inspired by a previous work by Filatrella et al. [5] for a three large generators grid model. We have based our analysis on the same assumptions but further introducing coupling perturbations under the form of square waves with different amplitudes in order to simulate a large coupling spread among nodes and mimic a sudden collapse of the couplings.
2 The Kuramoto model In the following, for the sake of clarity, we will use the following convention: matrices will be denoted by upper case boldface characters while vectors will be denoted by lower case boldface characters. Moreover, if A is a matrix, we will use the notation ai j to refer to its (i, j)th entry; likewise, if x is a vector, xi will denote its ith component. The standard Kuramoto model (SKM) [1], [2], [9] is a mean-field dynamic system resulting from a model of coupled oscillators whose phases are described to interact through a constant coupling as follows:
Stability of a distributed generation network using the Kuramoto models
K θ˙i = ωi + N
3
N
∑ sin(θ j − θi )
i = 1, . . . , N
(1)
j=1
where θi is the phase of the ith oscillator and depends on time t, ωi its natural frequency (natural frequencies are symmetrically distributed around ω0 ), K is strength of the constant coupling (same value for all links), N is the number of oscillators. The not-oriented oscillator network is supposed to be fully connected. In the case when lim (θi − θ j ) = 0
(2)
t→∞
oscillators synchronize, and their phase differences become asymptotically constant. Oscillators run independently at their natural frequencies, while couplings tend to synchronize them all, acting like a feedback. In order to measure the phase coherence, a so-called order parameter R has been introduced [2] K N iθi (3) R= ∑e N i=1 R ranges between 0 (no coherence) and 1 (perfect coherence). Kuramoto [2] showed that, for the model of (1), for K < kc (N −→ ∞) oscillators remain unsynchronized, while for K > kc they synchronize. If we modify SKM by introducing a generic adjacency matrix connecting nodes whose generic entry Ki j represents variable coupling strength between nodes and may vary as a function of time, we end up with a modified Kuramoto model equation which reads as follows: N
θ˙i = ωi + Km ∑ Ki j sin(θ j − θi )
i = 1, . . . , N
(4)
j=1
where Ki j values might be randomly selected and expressed as fractions of the maximum coupling value Km . The pertinence of the modified Kuramoto model to our purposes will be shown in the next section, where we derive the modified Kuramoto equation (4) starting from a Cardell-Ilic model of a distributetd network of power generators.
3 Derivation of the SKM from the Cardell-Ilic model The Cardell-Ilic model [6] is a linearized dynamic model for distributed generators (steam-turbine, combustion turbine, combined cycle, wind) in a power distribution system, using a very small number of state variables and incorporating the generated power as coupling variable among the individual models through the equation: ˜ + p˙ x˙ = Ax − Kω
(5)
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Vincenzo Fioriti, Silvia Ruzzante, Elisa Castorini, Elena Marchei, Vittorio Rosato
˜ is derived from the where x is the state vector, A is defined as the system matrix, K jacobian matrix, ω is the generator frequency and p˙ the power output. For the i-th row: x˙i = ∑ ai j x j − ∑ k˜ i j ω j + p˙i j
i = 1, . . . , N
(6)
j
only for the the state variables regarding the phase. Setting θ˙i = xi , p˙i = −
Ω0 max p sin(θ j − θi ) Ii i ∑ j
(7) i = 1, . . . , N
(8)
where Ii is the inertial moment, Ω0 the nominal system frequency and pmax is the i maximum power of the ith generator. We derive that [5]:
∑ ai j θ˙ j = ∑ k˜ i j ω j + j
j
Ω0 max p sin(θ j − θi ) Ii i ∑ j
i = 1, . . . , N
(9)
in which we recognise the same formal equation of Kuramoto model where at the left hand side we find a linear combination of θ˙ j , and at the right hand side a sum of a linear combination of frequencies and sinusoidal terms. Therefore, the simple, linearized Cardell-Ilic model is in relation with the SKM by means of the power couplings and expressing phases as linear combinations of the state variables of the complete distributed generation system (7). Many technical details have been neglected, but the general sense is that SKM can be used to map the dynamics of a distributed generation system.
4 The simulation of the modified SKM Fig. 1 (right) reports a sketch of a model of a distributed network composed by the connection of smaller subnetworks. Each node represents a power generating unit; they are connected in a ring topology. The choice of this specific topology is motivated by a recent finding [7] that synchronization is preserved in the generic case of a graph formed by connected ring subgraphs, as in the left side of Fig. 1. While their [7] demonstration refers to a wider topology class, we study a simple ring topology for a single block; results could be then generalized to a more general structure. Simulations have been thus carried out by using the network in Fig. 1 (right). Scope of the simulations is to measure how a time-dependent coupling between nodes might affect the system’s synchronization. To this purpose, we have introduced time-dependent ki j , under the form of square waves with amplitudes chosen from an uniform distribution between 0 and Km , and period from 55 min to 0.25 s, in order to simulate an abrupt change in the power. In fact, relevant problems for
Stability of a distributed generation network using the Kuramoto models
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Fig. 1 On the left side the block topology of the oscillators /generators, with the dissipating node. On the right side, a single block. This is the network used for the simulations whose results are the object of the present work.
DG stability are the dropped generators and the small inertia of the generators. Both these problems induce a frequency destabilization. The physical constraint of the energy conservation has been taken into account by considering a dissipation node (the black node in Fig. 1). A stringent quality of service has been asked for defining the onset of synchronization among nodes, by requiring a value of R as large as R > 0.8. Under this assumption, the critical value for the onset of system’s synchronization results to be Km ' 0.1, which will be retained as the critical threshold of the Kuramoto model kc . Below this threshold, synchronization does not take place.
5 Results The network (right side of Fig. 1) has been simulated using t = 104 s with steps of dt = 0.05 s Figs. 2, 3, 4 show the behavior of the phase angle θi and the order parameter R in case of low Km = 0.1 (Figs. 2, 3) and high Km = 400 (Figs. 4, 5) coupling cases, respectively.
Fig. 2 Phases with low coupling (Km = 0.1).
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Vincenzo Fioriti, Silvia Ruzzante, Elisa Castorini, Elena Marchei, Vittorio Rosato
Fig. 3 Order parameter R with low coupling (Km = 0.1).
As a general feature of the model when the maximum coupling strength Km is low, R behaves erratically (Fig. 3), while when it increases, R rapidly goes to 1, although several crises can be observed. Phase differences θi remain almost constant (see Fig. 6). In particular: • low coupling strength: in Fig. 3 the order parameter oscillates erratically around a mean value (different from zero) because Km is close to the critical value kc . Unfortunately, this is not sufficient to guarantee a sufficiently stable synchronization as < R > 0.8. • high coupling strength: the order parameter is 1 for most of the time; some deep “crises” are observed, but the system quickly recovers stability.
Fig. 4 Phases with high coupling (Km = 400).
In Fig. 5 Km = 400: as a result, R oscillates around the unity. Figs. 6, 7 show successive enlargements of the phase angle behaviour during the crisis of synchronization loss at Km = 400. It is relevant to observe that, during the synchronization losses, the phase angles tends to remain synchronized, although the spread between the phase angle grows (see Figs. 8, 9). As a further finding, we have also studied the rising time (i.e. the time for R to pass from zero to the unity value, see Fig. 10) as a function of the value of Km , follows a power low pattern (Fig. 11). Moreno and Pacheco [8] found that the resynchronization time of a perturbed node decays as a power law of its degree, for the SKM
Stability of a distributed generation network using the Kuramoto models
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Fig. 5 Order parameter R with high coupling (Km = 400).
Fig. 6 Enlargement of Fig. 5, t > 5000 s, high coupling.
Fig. 7 Enlargement of Fig. 5, (between 3500 and 4000 s).
in a scale free topology. Although we consider a simple ring topology, nevertheless the occurrence of two power laws may be clues of some kind of a self-organizing criticality (SOC) working in the KM. On the other hand, Carreras et al. [4] have suggested the SOC in power grids as an explanation to the blackouts. The meaning of Fig. 9 is that the restarting of a grid after a failure or during a control action, if an high value of the couplings is present, this will determine a fast re-synchronization, coping with the problem of the fault clearing time.
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Vincenzo Fioriti, Silvia Ruzzante, Elisa Castorini, Elena Marchei, Vittorio Rosato
Fig. 8 Phases synchronization crisis (enlargement between 3500 and 4000 s) for Km = 400.
Fig. 9 Further enlargements of the data shown in Fig.6.
Fig. 10 The rising time, for Km = 4.
Stability of a distributed generation network using the Kuramoto models
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Fig. 11 The power law: rising-time ts vs. max. coupling amplitude Km (log-log plot).
6 Conclusions We discuss some stability issue of a DG power system modelled through a small (seven nodes) network; the dynamics of the generator’s phase angles have been described using a modified Kuramoto model [10] derived by a Cardell-Ilic model [6] by using the interaction scheme proposed by Filatrella [5]. Differently from that efforts, in our model, internode couplings are allowed to vary as square waves, with randomly chosen coupling amplitudes. Under these assumptions, we observe the system, even for average coupling values beyond the instability value (Km > kc ) undergoing to several synchronization losses from which the system has been able, however, to quickly recovering. We have also shown that the rising-time of synchronization follows a power-law, in qualitative agreement with previously reported findings [8], [4]. Our results are also in agreement with recent findings of Popovych et al. [3]. They showed that, for N ≥ 4 and Km sub-critical, the SKM shows phase chaos as N increases, developing rapidly high-dimensional chaos for N = 10 with the largest Lyapunov exponent (LLE) at its maximum positive value. Then the LLE decreases very fast as 1/N, indicating a less chaotic regime. They conclude that, for an intermediate size system (in term of number of oscillators), a more intense phase chaos than small (N < 4) or large (N > 20) ones can be generated; our simulation [10] seems to confirm their conclusions (see Figs. 2, 3). Thought their results have been obtained for the standard SKM (i.e. one fixed K) they seem to support the idea that an intermediate (5 < N < 20) value for N should be avoided, in order to have a robust phase-lock. In conclusion, the modified Kuramoto model seems able to describe a distributed generation and various model instabilities, both in power amplitude and frequency. Some useful indications can be derived: coupling strength must be kept as high as possible which means high voltage transmission/distribution lines, DG size (number of nodes) should be very small or very large, the grid must ensure the coupling feedback actions by means of an appropriate topology. Simulation of the modified SKM with larger and more complex network topologies are planned.
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Vincenzo Fioriti, Silvia Ruzzante, Elisa Castorini, Elena Marchei, Vittorio Rosato
7 Acknowledgements The authors acknowledge fruitful discussion with R.Setola (Campus Biomedico). One of us (S.R.) acknowledges project CRESCO (PON 2000-2006, Misura II.2.a) for funding.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
H. Chia, Y. Ueda, Kuramoto Oscillators, Chaos Solitons & Fractals,12, 159, (2001). Y. Kuramoto, Chemical Oscillation, Springer, Berlin, (1984). O. Popovych et al., Phase Chaos in Coupled Oscillators, Phy. Rev. E, 71, 06520, (2005). B. Carreras et al., Evidence for SOC in a Time Series of Electric Power System Blackouts, Chaos, 51, 1733 (2004). G. Filatrella et al., Analysis of Power Grids using the Kuramoto Model, Eur. Phy. J. B, 61, 485, (2008). J. Cardell, M. Ilic, Maintaining Stability with Distribute Generation, Power Eng. Soc. Meeting IEEE, (2004). E. Canale, P. Monzon, Gluing Kuramoto Coupled Oscillators Networks, IEEE Decision and Control Conf., New Orleans, (2007). Y. Moreno, A. Pacheco, Synchronization of Kuramoto Oscillators in Scale-Free Networks, Europhys. Lett., 68, (4), 603, (2004). J. Acebron et al., The Kuramoto Model, Rew. Mod. Phy., 77, 137, (2005). V. Fioriti, V. Rosato, R. Setola, Chaos and Synchronization in Variable Coupling Kuramoto oscillators, Experimental Chaos Catania, (2008). http://www.iset.uni-kassel.de/publication/2007/2007 Power Tech Paper.pdf. J. Carsten et al., Riso Energy Report, (2000). J. Cardell, M. Ilic, The Control of Distributed Generation, Kluver Academic Press, (1998).
