Synchronization transition of random model networks

For the large model networks of random, small-world and regular grids we introduce the order parameter r(t) to quantify the degree of synchronization. The order parameter [30]

is now defined as

r(t) = 1 N

N

X

j=1

e^iθj(t). (6.6)

If the system is fully synchronized, i.e., all phases are equal, the real part of the order parameter is <(r(t)) = 1. In a phase-locked state the real part of the order parameterr(t) can in general be any positive value between zero and one. Here <(r(t)) is in fact close to one for all of our model networks if the system is in a phase-locked state. If the system does not reach a phase-locked state, the real part of the order parameter fluctuates around zero.

An example for the relation between the phases of the machines and the order parameter is illustrated in Figure 6.4. The dynamics of the phasesθ_j(t) of all machinesj and the real part of the order parameter are shown for two different values of the coupling strengthK. Without coupling,K = 0, all elements of the grid oscillate with their natural frequency ω.

For small values of K with K < K_c, only the phases of the decentralized generators and the consumers are close together as illustrated in Figure 6.4(a). The system is thus not in its stable state and the real part of the order parameter fluctuates around zero. If the coupling strength is further increased, as illustrated in Figure 6.4(b), such that we have K > K_c, all generators phase-lock as well, such that a stable operation of the power grid is possible. Consequently, the real part of the order parameter is close to one.

In the long time limit, the system will either relax to a steady phase-locked state or to

6. Decentralized power generation in future power grids

Figure 6.4: Phase dynamics of a quasi-regular power grid. (a) For weak coupling the phases θ_j(t) of the small renewable decentralized generators (green lines) are close to the consumer’s phases (blue lines), but not the phases of the large power plants (red lines). Thus the order parameter r(t) fluctuates around a zero mean. (b) Global phase-locking of all generators and consumers is achieved for a large coupling strength, such that the real part of the order parameters r(t) has a positive value (here close to one).

a limit cycle where the generators and consumers are decoupled and <(r(t)) oscillates around zero. In order to quantify synchronization in the long time limit we thus define the averaged order parameter

In numerical simulations the integration time t₂ must be finite, but large compared to the oscillation period if the system converges to a limit cycle. Furthermore, we consider the averaged squared phase velocity

as a measure of whether the grid relaxes to a steady state or not. In the steady state we havev∞= 0 because all phase derivatives are zero (cf. section5.2). If we havev∞6= 0, the system is not in its stable state. The two quantitiesr∞and v∞ are plotted in Figure 6.5 as a function of the coupling strengthK/P₀ for 20 realizations of a quasi-regular network with N_C = 100 consumers and 40% renewable energy sources. The onset of phase-locking is

52

6.2 The synchronization transition

0 2 4 6 8

0 0.5 1

K/P0

r !

0 2 4 6 8

0 20 40

K/P0

v !

Figure 6.5: The synchronization transition as a function of the coupling strengthK: The order parameterr∞(left-hand side) and the phase velocityv∞ (right-hand side) in the long time limit.

The dynamics has been simulated for 20 different realizations of a quasi-regular network consisting of 100 consumers,N_P = 6 large power pants andN_R= 16 small power generators.

clearly visible: If the transmission capacity is smaller than the critical value K_c, no steady phase-locked state exists and we haver∞= 0. Increasing K aboveK_c leads to the onset of phase-locking such that r∞ jumps to a non-zero value. The critical value of the coupling strength is found to lie in the range K_c/P₀ ≈3.1−4.2 where v∞ reaches zero. The critical value depends on the random realization of the network topology.

The synchronization transition is quantitatively analyzed for the three different general network topologies in Figure 6.6. We plotted r∞ and v∞ averaged over 100 random re-alizations for each amount of decentralized energy sources and for every topology. The synchronization transition strongly depends on the structure of the network and in par-ticular the amount of power provided by small decentralized energy sources. Each line in Figure 6.4 corresponds to a different fraction of decentralized energy 1−N_P/10, whereN_P is the number of large conventional power plants feeding the grid. Most interestingly, the introduction of small decentralized power sources (i.e. the reduction of N_P) promotes the onset of phase-locking. The onset of phase-locking is most obvious for the random and the small-world structures as illustrated in Figure 6.6(a,b).

Let us analyze the quasi-regular grid in the limiting cases N_P = 10 (only large power plants) and N_P = 0 (only small decentralized power stations) in detail. The existence of a phase-locked steady state requires that the transmission lines leading away from a generator have enough capacity to transfer the whole power, i.e., 10P₀ for a large power plant and 2.5P₀ for a small power station. In a quasi-regular grid every generator is connected with exactly four transmission lines, which leads to the following estimate for the critical coupling strength (cf. (6.5)):

K_c= 10P₀/4 forN_P = 10, (6.10a)

K_c= 2.5P₀/4 forN_P = 0. (6.10b)

These values only hold for a completely homogeneous distribution of the power load and thus rather present a lower bound for K_c in a random network realization. Indeed, the

6. Decentralized power generation in future power grids

Fraction of distributed energy sources [%]:

0 2 4 6 8

Figure 6.6: The synchronization transition for different fractions of decentralized energy sources 1−N_P/10 feeding the grid and for different network topologies: (a) Quasi-regular grid, (b) random network and (c) small-world network. The order parameter r∞ and the phase velocity v∞ have been averaged over 100 realizations for each network structure and each fraction of decentralized sources.

numerical results illustrated in Figure 6.6(a) yield a critical coupling strength of K_c ≈ 3.2×P₀andK_c≈1×P₀, respectively (cf. (6.5) and (6.2)). However, the motifs provide only rough estimates and may serve as lower bounds for the actual synchronization transition because topological disorder typically increases the synchronization threshold [68].