Distributed adaptive clock synchronization scheme for smart grid based on system stability analysis

Abstract

Aiming at services such as nearby consumption and flexible interaction of new loads, it is necessary to achieve higher precision clock synchronization of local equipment, and the public network transmission delay fluctuates randomly, which cannot meet the corresponding accuracy requirements. In order to get rid of the limitations brought by public network transmission, distributed clock synchronization replaces the centralized timing of the master station through information exchange between nodes to achieve higher precision clock synchronization. However, in the process of distributed synchronization, a large amount of information will be exchanged between key nodes of the link and their neighbors, but the concurrent transmission capacity of key nodes such as dedicated power gateways and centralized coordinators (CCOs) is limited, which is difficult to meet the corresponding communication requirements. In order to solve the above problems, this paper proposes a time-triggered distributed synchronization scheme based on node priority, which can meet the requirements of high-precision synchronization and reduce the transmission burden of link aggregation nodes such as power dedicated gateways and CCOs.

Keywords

smart grid distributed clock synchronization node priority event-triggered mechanism

Introduction

Public network information transmission experiences significant delay fluctuations, making it challenging to meet the high-precision synchronization requirements of certain new power applications through centralized time distribution. To mitigate the impact of public network transmission delays on synchronization accuracy, local synchronization via clock information exchange among distributed devices is necessary. In distributed synchronization, the absence of a centralized reference clock necessitates information exchange between nodes to establish a standard reference time, which may deviate from standard coordinated time. The basic implementation process of distributed synchronization involves the following steps: based on the clock data packets broadcast by neighboring devices, user devices compare their own clock state with that of neighboring devices and further adjust their clocks. Currently, clock adjustment in distributed synchronization involves establishing dynamic equations for nodes,^1,2 determining the node clock as a function of its current clock state and neighboring clock states. Parameters compensation values for the equations are obtained through the exchange of clock information between nodes, enabling distributed synchronization in wireless network scenarios.³ Therefore, frequent exchanges of clock information between neighboring nodes occur, with many critical link neighbors and high pressure from concurrent data packet transmissions.⁴ In power networks, not only is network bandwidth capacity limited, but the concurrent transmission capability of critical nodes such as power-specific gateways and CCOs is also constrained, making it difficult to achieve concurrent data packet exchanges among hundreds to thousands of devices in a distributed network.^5,6 Hence, optimization of the information exchange process in distributed synchronization for power networks is necessary.

In traditional sensor networks, a reference node is set up and synchronization is gradually achieved around this central reference node, which still places a significant transmission burden on the central node.⁷ Shi et al. proposed a rapid flood-based multicast clock synchronization method with real-time delay compensation,⁸ which significantly improves synchronization accuracy by estimating cumulative errors. However, this method incurs high network resource consumption due to broadcasting floods after estimating errors through bidirectional time exchanges. For distributed system synchronization in Wi-Fi networks, Chen et al. proposed a PTP synchronization method based on a combination of software and hardware timestamps, implemented on x86 and ARM hardware platforms. Nevertheless, this method still requires substantial information exchange.⁹ Yan et al. applied Graph Neural Networks to the resilient synchronization of aerial swarms with low communication overhead.¹⁰ However, it requires strong computing power of terminal devices, which does not match the power scenario.

In order to reduce the burden on central nodes, distributed cooperative synchronization schemes do not require a reference node. Therefore, all nodes achieve clock synchronization of the whole network through peer-to-peer interaction with neighboring nodes in existing methods. For example, Yildirim et al. utilize an adaptive proportional-integral controller to compensate for deviations between different clocks.¹¹ However, the method of adjusting constants under specific conditions does not effectively address the deviations. Tian et al. employ an undirected graph network model to address multi-agent synchronization issues with communication delays,¹² but this approach may still result in steady-state errors. Parvez et al. proposed a synchronization scheme based on Gossip algorithms.¹³ To reduce bandwidth usage, this scheme randomly exchanges information between only two users at a time, which leads to slow convergence of the algorithm.

In practical implementations of synchronization schemes, continuous exchange of information between nodes is required. However, the capacity of power networks is highly constrained, and random fluctuations in transmission delays due to network bandwidth limitations are common, affecting the convergence of algorithms.^14,15 As a result, some researchers have explored event-triggered mechanisms to reduce interaction frequency. Dimarogonas et al. proposed a distributed event-triggered control mechanism with a single integrator but only considered state-based triggering without accounting for time-based triggers.¹⁶ Kadowaki et al., based on event-triggered mechanisms,¹⁷ designed a clock synchronization mechanism for sensor networks. Nonetheless, their approach still requires a reference node, which incurs significant transmission pressure on the reference node, making it unsuitable for power distribution networks. Andras et al. purposed a novel synchronized system based on a dithering algorithm to improve the frequency resolution of the integrated phase-locked loop (PLL), as the frequency resolution of these integrated PLLs is found to be to limiting factor.¹⁸ However, this type of algorithm is highly hardware depended leading to the high cost of equipment.

With a vast number of devices and multiple services coexisting, the actual power network faces large volumes of transmitted information and complex topologies, resulting in significant delays in public network transmission. Existing works have not addressed the high-precision synchronization requirements for device interactions in the context of long public network delays. This paper proposes a time-triggered distributed synchronization scheme based on node priority, which meets high-precision synchronization needs while reducing the transmission burden on aggregation nodes such as power-specific gateways and CCOs.

Distributed clock synchronization model

Distributed smart grid model

In the context of renewable energy power networks, two synchronization mechanisms defined by IEEE 1588 protocol—End-to-End (E2E) and Peer-to-Peer (P2P)—coexist. These mechanisms can be represented using a directed graph $G = G (Q, U, A)$ , where nodes $Q = {1, 2, \dots, N}$ denote the renewable devices to be synchronized. The set of directed edges $U \subseteq Q \times Q$ represents the communication links between these devices. The adjacency matrix $A = [a_{i j}] \in R^{N \times N}$ shows the state of the communication links.

For each user device in the distributed network, it is assumed that a nonlinear system dynamic equation is used to represent the current clock state and its offset, as well as the clock inputs from neighboring nodes. This can be formulated as:

{\dot{x}}_{i} = A x_{i} + f (x_{i}, t) + B u_{i},

(1)

where

x_{i}

represent the current clock state of the user device,

u_{i}

denote the input clock information at the current time,

A

be the weighted adjacency matrix of the graph, and

B

be the input equation of the system, which describes the influence of the input on the system’s state. This can be considered as a clock offset model.

The $x_{i}$ employs a 4D state indicating to: (1) Clock Phase Offset (sec): Absolute time difference from reference (e.g., UTC). (2) Clock Frequency Offset (Hz): Deviation of oscillator frequency from nominal rate. (3) Phase Rate of Change (sec/s): Rate of phase drift (critical for instability mitigation). (4) Environmental Disturbance: Models external factors (e.g., temperature-induced drift).

Assume that the nonlinear mapping $f (x_{i}, t)$ satisfies the Lipschitz condition, meaning $f : R^{N} \times R_{\geq 0} \to R^{N}$ is piecewise continuous and there exists a Lipschitz constant $L \in R_{\geq 0}$ such that for any $y, z \in R^{N},$ $‖ f (y, t) - f (z, t) ‖ \leq L ‖ y - z ‖ .$ The Lipschitz constant provides an upper bound on the rate of change of the mapping $f$ . A smaller $L$ implies that the function exhibits less variation in output with respect to changes in input, indicating that the function is smoother.

Assume that the graph $G$ is strongly connected. Due to the communication constraints in the power grid, each device can only receive clock information from its neighboring devices. For a user device i, let the input be represented as $u_{i} = u_{i} (x_{i}, x_{j}, t)$ , where j is the predecessor neighbor of i, i.e. $j \in N_{i}^{i n}$ . The goal of distributed clock synchronization is to determine the specific form of the input function such that the clock deviations of the power devices asymptotically approach zero, meaning that as $t \to \infty$ , $x_{1} (t) = x_{2} (t) = \dots = x_{N} (t)$ . Moreover, for all user devices $i \in Q$ , as $t \to \infty$ , the dynamic equation ${\dot{x}}_{i} = A x_{i} + f (x_{i}, t)$ can be determined, and at this point, the input $u_{i} (t) \to 0$ ensures that the system reaches a synchronized state.

Distributed clock error adaptive adjustment scheme

The event-triggered synchronization mechanism

To meet the stringent communication bandwidth constraints of power networks and reduce the number of data packet exchanges between user devices, this section proposes a design based on an event-triggered mechanism. Let ${\hat{x}}_{j}^{i}$ denote the estimated clock of neighbor device j by user device i. The input clock information in the system’s dynamic equations can be expressed using the estimated clocks of neighbor nodes, as showed in equation referring to 1.

u_{i} = c K \sum_{j \in N_{i}^{i n}} a_{i j} (x_{i} - {\hat{x}}_{j}^{i}) .

(2)

when the estimate of

{\hat{x}}_{j}^{i}

is sufficiently accurate, the distributed network does not require continuous information exchanges to achieve synchronization between user devices. The estimation method for

{\hat{x}}_{j}^{i}

can be expressed as equation (3).

{\hat{x}}_{j}^{i} : {\begin{cases} {\dot{\hat{x}}}_{j}^{i} = A {\hat{x}}_{j}^{i} + f ({\hat{x}}_{j}^{i}, t) t \in [t_{i j, k}, t_{i j, k + 1}) \\ {\hat{x}}_{j}^{i} (t_{i j, k}^{+}) = x_{j} (t_{j, k}) \end{cases},

(3)

where j is the predecessor neighbor of i,

{t_{j, k}}

is the time when user device j broadcasts time information and

{t_{j, k}}

is the time when device i receives the broadcast information from j. When high-precision synchronization between devices is achieved, the input

u_{i}

is zero, and the estimate reflects the current state of the user device, thus satisfying equation (3). To further compare the estimated value

{\hat{x}}_{j}^{i}

and the true clock

x_{j}

, the deviation can be expressed as:

{\hat{x}}_{j} : {\begin{cases} {\dot{\hat{x}}}_{j} = A {\hat{x}}_{j} + f ({\hat{x}}_{j}, t) t \in [t_{j, k}, t_{j, k + 1}) \\ {\hat{x}}_{j} (t_{j, k}^{+}) = x_{j} (t_{j, k}) \end{cases} .

(4)

Assuming that communication delays between user devices can be neglected, it follows that after device j broadcasts its clock information (for all successor neighbors k of device j). The remaining neighbors receive it immediately, as all devices now share identical information about j’s state. In the model, we have further assumed that the global estimate inherits the same dynamic structure as the local estimate. In this case, the input can be represented by equation (5).

u_{i} (t) = c K \sum_{j \in N_{i}^{i n}} a_{i j} (x_{i} - {\hat{x}}_{j}) .

(5)

It is crucial to find an appropriate pair of $c$ and $K$ needs to be selected to achieve system synchronization. For the case where user device j is not a predecessor neighbor of user device i, $a_{i j} = 0$ can be transformed to equation (6):

u_{i} (t) = c K \sum_{j = 1}^{N} a_{i j} [x_{i} - x_{j} + (x_{j} - {\hat{x}}_{j})] .

(6)

For each user j, there is a deviation between the true clock value and the estimated value from other devices, which can be expressed as equation (7):

e_{j} = (x_{j} - {\hat{x}}_{j}) .

(7)

If the estimation deviation of a user towards its neighboring users is bounded, the system synchronization error will also be bounded. When $‖ e_{j} ‖$ exceeds a pre-set threshold $h_{j}^{'} (t)$ , an event mechanism is triggered. User j broadcasts its clock to neighbors, and the event trigger function is given by equation (8):

δ_{j} (t) = ‖ e_{j} (t) ‖ - h_{j}^{'} (t) .

(8)

Clock synchronization strategy based on node priority

In practical power networks, the concurrent transmission capability of critical nodes in links, such as CCO, is constrained, making it challenging to meet the demands for frequent clock information exchanges with all nodes. To further alleviate the information transmission burden on critical nodes in links, a threshold function design based on node priority has been proposed, as illustrated in equation (9).

h_{i} = h_{i}^{'} \times D^{i n},

(9)

where

h_{i}

is the revised threshold function and

D^{i n}

represents the in-degree of a node. To quantitatively describe the synchronization frequency of key nodes, a new evaluation method based on the average update frequency of nodes is proposed. Nodes with an in-degree of 1 are on the periphery of the network, thus the communication cost for synchronization is negligible. For critical nodes with a degree greater than 1 in the topology, the average number of clock changes from the initial state to the converged equilibrium state is used for description. The expression for this metric is given by equation (10). A smaller value of this expression indicates fewer clock broadcasts by critical nodes in the system, implying lower parallel transmission pressure.

F_{s y n} = \frac{1}{n^{'}} \sum_{i = 1, D_{i}^{i n} > 1}^{n} D_{i}^{i n} \times F_{i} .

(10)

where

F_{i}

denotes the number of updates for node i and

n^{'}

represents the number of nodes with in-degrees greater than 1. The pseudocode for the calculation is as follow:

Distributed clock error adaptive adjustment scheme

The synchronization error of user devices can be described using the clock deviation between user device ii and the other user devices. This is represented by equation (11):

ξ_{i} = x_{i} - \sum_{j = 1}^{N} r_{j} x_{j},

(11)

where

r_{j}

is the normalized representation of the in-degree of vertex j, reflecting its importance in the graph. Essentially,

r_{j}

is the j-th component of the left eigenvector of the Laplacian matrix. If the graph is connected and balanced, then

r = 1 / N

. Let

ξ = {ξ_{1}^{T}, \dots, ξ_{N}^{T}}

be the synchronization error matrix, which can be used to describe the system’s synchronization state. This is represented by equation (12):

ξ = Wx,

(12)

where

x = {x_{1}^{T}, \dots, x_{N}^{T}}

I_{N}

is the identity matrix of dimension equal to the number of users,

I_{n}

is the identity matrix of dimension equal to the number of user states, and

W

is the matrix describing the connectivity among users. The pseudocode for calculation of

c

and

K

is as Table 1.

Table 1.

Event triggering mechanism algorithm.

Algorithm 1: Event triggering algorithm (example for node i)
1: At each time step, user i performs the following steps:
2: function Communication and Coordination
3: Adjust the threshold function based on node priority and calculate.
4: if the triggering function is greater than 0, then:
5: Broadcast the clock information of user i;
6: Recalculate ${\hat{x}}_{j}$ ;
7: end if
8: if a new message is received from user j then
9: if j is a predecessor neighbor of i:
10: recalculate ${\hat{x}}_{j}^{i}$ ;
11: end if
12: end if
13: recalculate $u_{i}$ ;
14: end

The matrix can be represented as equation (13) if and only if all devices achieve synchronization.

W = (I_{N} - 1 \cdot r^{T}) \otimes I_{n} .

(13)

Linear Matrix Inequalities (LMIs) are used to ensure that a positive definite matrix $P$ is found, such that all conditions for Lyapunov stability of the closed-loop system are satisfied. The computation of parameters $c$ requires global topological information of the power communication network, which is determined based on the connectivity of nodes and the network’s topology. In contrast, $K$ only requires computation based on the state of the power devices themselves. For simplicity in proving convergence, it is assumed that all network topologies are globally known.

Let γ be the vector function containing all thresholds. Assume that a solution to the LMI problem in Algorithm 2 exists, that is, the algorithm converges. It is necessary to prove that the following three theorems hold:

1. Theorem 1: There exist a K-type function $γ$ and a KL-type function $β$ such that for any input state $ξ (t_{0})$ , the system synchronization error $ξ$ satisfies equation (15). Where a K-type function $γ$ is defined as strictly increasing on the positive semi-axis and $γ (0) = 0$ ; a KL-type function $β (s, r)$ is defined as decreasing in s when r is fixed and $\lim_{s \to \infty} β (r, s) \to \infty$ . The physical meaning is that for any initial state, the synchronization error $ξ$ satisfies certain boundary conditions. This means that the synchronization error is within a range centered around the origin, and this range can be reduced by adjusting system parameters.

∥ ξ (t) ∥ \leq β (‖ ξ (t_{0}) ‖, t - t_{0}) + γ (\sup_{t_{0} \leq τ \leq t} ∥ h (t) ∥) .

(15)

2. Theorem 2: For N user devices, if the threshold function is bounded and represented as $\lim_{t \to \infty} h_{i} (t) = c_{l}$ , then equation (16) holds. Theorem 2 indicates that if the triggering function is a bounded function by design, the synchronization error can be limited to the lower range which is determined by the bound of the triggering function.

\begin{array}{l} \lim_{t \to \infty} ∥ ξ (t) ∥ \leq r \\ = \frac{2 ∥ F_{1} ∥ λ_{\max} (R \otimes P^{- 1})}{λ_{\min} (H_{2}) λ_{\min} (R \otimes P^{- 1})} \sqrt{N} c_{l}, \end{array}

(16)

where

F_{1} = (R \otimes P^{- 1}) W (L \otimes c B B^{T} P^{- 1}),

(17)

H_{2} = - (I_{N} \otimes P) (R \otimes H_{1}) (I_{N} \otimes P),

(18)

H_{1} = AP + P A^{T} - 2 τ B B^{T} + μ l I_{n} + μ^{- 1} l P^{2},

(19)

where

P

is a positive definite matrix obtained from solving a linear matrix inequality in algorithm 2,

τ

and

μ

are the parameters obtained.

R

represents the left eigenvector of the Laplacian matrix,

W

denotes the clock comparison function between the user and neighbor nodes.

L

is the Laplacian matrix, l is the corresponding Lipschitz constant.

λ_{\max}

and

λ_{\min}

are the maximum and minimum eigenvalues of matrix

H_{2}

, respectively.

3. Theorem 3: If the threshold function $h_{i} (t)$ has a lower bound $c_{l} > 0$ , then all user devices can converge to a synchronized state, and the system does not exhibit Zeno behavior. Zeno behavior refers to the phenomenon where the threshold function is greater than the synchronization error, leading to continuous synchronization triggers but never achieving the required synchronization precision.

Proof of Theorem 1

To prove the existence of Theorem 1, substitute the input expressions, and the synchronization mechanism’s expression becomes (20).

\dot{x} = (I_{N} \otimes A + c L \otimes BK) x + g (x, t) + η,

(20)

where

g (x, t) ≜ [\begin{array}{c} f (x_{1}, t) \\ ⋮ \\ f (x_{N}, t) \end{array}], η ≜ [\begin{array}{c} c BK \sum_{j = 1}^{N} a_{1 j} e_{j} \\ ⋮ \\ c BK \sum_{j = 1}^{N} a_{N j} e_{j} \end{array}] .

(21)

Thus, the synchronization error is expressed as (22).

\begin{array}{l} \dot{ξ} = W \dot{x} = (I_{N} \otimes A + c L \otimes BK) ξ \\ + W (g (x, t) + η) . \end{array}

(22)

A Lyapunov function can be defined as (23).

V = ξ^{T} (R \otimes P^{- 1}) ξ .

(23)

Since $P ≻ 0$ and $R$ is a positive diagonal matrix, $V$ is evidently positive definite, and the time derivative is given by (24).

\begin{array}{l} V = \underset{= X}{\underset{⏟}{2 ξ^{T} (R \otimes P^{- 1}) (I_{N} \otimes A + c L \otimes BK) ξ}} \\ \begin{array}{l} + \underset{= Y}{\underset{⏟}{2 ξ^{T} (R \otimes P^{- 1}) Wg (x, t)}} \\ + \underset{= Z}{\underset{⏟}{2 ξ^{T} (R \otimes P^{- 1}) W η}} \end{array} \end{array},

(24)

where

X

represents the linear interaction between agents affecting synchronization error,

Y

represents the nonlinear function affecting synchronization error, and

Z

reflects the impact of the event-triggered mechanism on the system’s synchronization error. Together, these factors determine the system’s synchronization performance and stability. Let

ζ = (I_{N} \otimes P^{- 1}) ξ

and substitute the control parameters

K = - B^{T} P^{- 1}

from Algorithm 2. Since

(r^{T} \otimes I_{n}) ξ = 0

and

(r^{T} \otimes I_{n}) ζ = 0

, according to Theorem 4.9 in 11, (25) can be obtained.

ζ^{T} (c (RL + L^{T} R) \otimes B B^{T}) ξ \geq 2 ξ^{T} (R \otimes c a (L) B B^{T}) ξ .

(25)

Thus, the upper bound of $X$ can be represented as (26).

X \leq ζ^{T} (R \otimes (AP + P A^{T} - 2 τ B B^{T})) ζ .

(26)

Let $\bar{x} = \sum_{i = 1}^{N} r_{i} x_{i}$ , where $r_{i}$ is the i-th component of the left eigenvector $R$ of the Laplacian matrix, and it can be expressed as (27).

\begin{array}{l} Y = 2 \sum_{i = 1}^{N} r_{i} ξ_{i}^{T} P^{- 1} (f (x_{i}, t) - \sum_{j = 1}^{N} r_{j} f (x_{j}, t)) \\ \begin{array}{l} = \underset{= Y_{1}}{\underset{⏟}{2 \sum_{i = 1}^{N} r_{i} ξ_{i}^{T} P^{- 1} (f (x_{i}, t) - f (\bar{x}, t))}} \\ + \underset{= Y_{2}}{\underset{⏟}{2 \sum_{i = 1}^{N} r_{i} ξ_{i}^{T} P^{- 1} (f (\bar{x}, t) - \sum_{j = 1}^{N} r_{j} f (x_{j}, t)) .}} \end{array} \end{array}

(27)

Since $\sum_{i = 1}^{N} r_{i} ξ_{i} = 0$ , it follows that $Y_{2} = 0$ . Using the properties of Lipschitz functions and Young’s inequality, the expression for $Y$ is obtained as (28).

\begin{array}{l} \begin{array}{l} Y = Y_{1} \leq 2 \sum_{i = 1}^{N} r_{i} l ξ_{i}^{T} P^{- 1} ξ_{i} \\ \leq \sum_{i = 1}^{N} r_{i} l ξ_{i}^{T} (μ {(P^{- 1})}^{2} + μ^{- 1} I_{n}) ξ_{i} \end{array} \\ = ζ^{T} (R \otimes (μ l I_{n} + μ^{- 1} l P^{2})) ζ \end{array} .

(28)

Next, we need to compute the upper bound for $Z$ . Define $e = {[e_{1}^{T}, \dots, e_{N}^{T}]}^{T}$ , then $η$ is given by (29).

η = (A \otimes c BK) e = - (A \otimes c B B^{T} P^{- 1}) e .

(29)

The upper bound of $Z$ can be obtained as (30).

\begin{array}{l} Z \leq 2 ‖ ξ ‖ ‖ (R \otimes P^{- 1}) (A \otimes c B B^{T} P^{- 1}) ‖ ‖ e ‖ \\ = 2 ‖ ξ ‖ ‖ F_{1} ‖ ‖ e ‖ \end{array} .

(30)

The upper bound of $V$ ’s time derivative can be represented as (31).

V \leq ζ^{T} (R \otimes H_{1}) ζ + 2 ‖ ξ ‖ ‖ F_{1} ‖ ‖ e ‖,

(31)

where

H_{1}

is a symmetric and negative definite matrix. Since

ζ = (I_{N} \otimes P^{- 1}) ξ

and

ξ = (I_{N} \otimes P) ζ

hold, the derivative of

V

can be transformed as (32).

\begin{array}{l} V \leq ζ^{T} (R \otimes H_{1}) ζ + 2 ‖ ξ ‖ ‖ F_{1} ‖ ‖ e ‖ \\ \begin{array}{l} \leq - λ_{\min} (H_{2}) {‖ ξ ‖}^{2} + 2 ‖ ξ ‖ ‖ F_{1} ‖ ‖ e ‖ \\ \leq - α_{ξ} (‖ ξ ‖), \forall ∥ ξ ∥ \geq ρ (‖ e ‖) \end{array} \end{array} .

(32)

where

α_{ξ} (s) = (1 - θ) λ_{\min} (H_{2}) s^{2}

and

ρ (s) = \frac{2 ‖ F_{1} ‖}{θ λ_{\min} (H_{2})}, θ \in (0, 1)

. The input-to-state Stability (ISS stability) of the synchronization error system is established in 18. ISS stability is a nonlinear control theory concept where a system’s state remains bounded when its inputs are bounded, and the state decays to zero when inputs vanish. Since the error

e_{i}

satisfies

‖ e_{i} (t) ‖ \leq h_{i} (t)

. For any

t \geq t_{0}

‖ e (t) ‖ \leq ‖ h (t) ‖

holds, thus proving Theorem 1.

Proof of Theorem 2

Define two K-class functions:

α_{1} (‖ ξ ‖) = λ_{\min} (R \otimes P^{- 1}) {‖ ξ ‖}^{2}

(33)

α_{2} (‖ ξ ‖) = λ_{\max} (R \otimes P^{- 1}) {‖ ξ ‖}^{2}

(34)

Clearly, the Lyapunov function satisfies $α_{1} (‖ ξ ‖) \leq V \leq α_{2} (‖ ξ ‖)$ for any state $ξ$ . Hence, (35) is obtained.

\begin{array}{l} γ (s) = α_{1}^{- 1} (α_{2} (ρ (r))) \\ = \frac{2 {‖ F ‖}_{1} λ_{\max} (R \otimes P^{- 1})}{θ λ_{\min} (H_{2}) λ_{\min} (R \otimes P^{- 1})} r . \end{array}

(35)

For all users $i \in Q$ , $\lim_{t \to \infty} h_{i} (t) = c_{l}$ , it follows that $\lim_{t \to \infty} ‖ h (t) ‖ = \sqrt{N} c_{l}$ . Substitute these relationships into the expression of Theorem 1, thus proving Theorem 2.

Proof of Theorem 3

To prove the existence of a minimum event interval when $c_{l} \geq 0 c_{l} \geq 0$ is positive, consider the continuous information sent by user i to neighbor nodes at times $t_{i, k}$ and $t_{i, k + 1}$ . The continuous error $e_{i} (t)$ under function $e_{i} (t) = x_{i} - {\hat{x}}_{i} = Ae + (f (x_{i}, t) - f ({\hat{x}}_{i}, t)) + B u_{i}$ and since ${\hat{x}}_{i} (t_{i, k}^{+}) = x_{i} (t_{i, k}^{+})$ , it follows that $e_{i} (t_{i, k}^{+}) = 0$ , and the upper bound for $‖ e_{i} (t) ‖$ is given by (36).

\begin{array}{l} \begin{array}{l} ‖ e_{i} (t) ‖ \leq \int_{t_{i, k}}^{t} ‖ A e_{i} (τ) ‖ d τ \\ + \int_{t_{i, k}}^{t} ‖ f (x_{i} (τ), τ) - f ({\hat{x}}_{i} (τ), τ) ‖ d τ \\ + \int_{t_{i, k}}^{t} ‖ B u_{i} (τ) ‖ d τ \end{array} \\ \leq \int_{t_{i, k}}^{t} (‖ A ‖ + l) ‖ e_{i} (τ) ‖ d τ \\ + \int_{t_{i, k}}^{t} ‖ B u_{i} (τ) ‖ d τ . \end{array}

(36)

The last formula is derived from the Lipschitz property of the function. To find the upper bound for $e_{i}$ , first determine the upper bound for $u_{i}$ . Let $u = {[u_{1}^{T}, \dots, u_{N}^{T}]}^{T}$ , then substituting the control parameters $K = - B^{T} P^{- 1}$ from Algorithm 2 yields the expression for $u$ .

\begin{array}{l} u = - (c L \otimes B B^{T} P^{- 1}) x - (A \otimes B B^{T} P^{- 1}) e \\ = - G ξ - Te \end{array},

(37)

where

G = c L \otimes B B^{T} P^{- 1}

T = (A \otimes B B^{T} P^{- 1})

, and for

i \in Q

, the upper bound of

u_{i}

is (38).

‖ u_{i} (t) ‖ \leq ‖ u (t) ‖ \leq ‖ G ‖ ‖ ξ (t) ‖ + ‖ T ‖ ‖ e (t) ‖ .

(38)

Since it has been previously proven that $‖ e (t) ‖ \leq ‖ h (t) ‖ \leq \sqrt{N} c_{u}$ always holds under condition t $\geq t_{0}$ , $c_{u}$ is the upper bound of $h_{i} (t)$ , and the upper bound of $e_{i} (t)$ is obtained as (39).

‖ e_{i} (t) ‖ \leq λ (t) + c \int_{t_{i, k}}^{t} λ (s) e^{c (t - s)} d s .

(39)

Applying the Gronwall–Bellman inequality, (40) is derived.

‖ e_{i} (t) ‖ \leq λ (t) + c \int_{t_{i, k}}^{t} λ (s) e^{c (t - s)} d s .

(40)

Since synchronization events are triggered only when $δ_{i} (t)$ is not zero or $‖ e_{i} (t) ‖ = h_{t} (t)$ and the time of the next event is not earlier than $t^{*} > t_{i, k}$ the minimum event time interval based on equation $Δ (t) = c_{l}$ is given by (41).

τ_{1} : t^{*} - t_{i, k} = l \frac{n (l + c_{l} (‖ A ‖ + l) / (‖ B ‖ \bar{u}))}{‖ A ‖ + l} > 0 .

(41)

Since there is a positive lower bound on the intervals between events, there are no accumulation points for synchronization errors in the sequence, and Zeno behavior does not occur. Thus, Theorem 3 is proved.

The proof of Theorem 1 shows that the synchronization error is bounded for any bounded threshold function, and in an event-triggered mechanism, the threshold function is used to adjust communication rates and synchronization performance. The result in Theorem 2 extends Theorem 1, indicating that if the threshold function converges, the synchronization error vector will also converge. That is, by reducing $c_{l}$ , the deviation can be made arbitrarily small. When $h_{i} (t) = 0$ , the algorithm degenerates to periodic synchronization.¹⁹

Simulation results analysis

Synchronization convergence analysis

In this section, MATLAB is used to numerically simulate the algorithm proposed earlier to verify the convergence of the distributed clock error adaptive adjustment scheme. First, the feasibility of the synchronization scheme is tested in a linear network scenario with three users. The whole procedures of the simulation can be conclude in four steps:

(1) Initialization: Initial clock states $x_{i} (0)$ randomized uniformly in [−1,1] to simulate real-world device heterogeneity.

(2) Event triggering: At each timestep, compute and broadcast the synchronization error $e_{i}$ , the vague $δ_{j}$ .

(3) Convergence Criteria: Simulation halted when $\max ‖ x_{i} - x_{j} ‖ < 0.1 m s$ for each $i, j$ , aligning with smart grid sync standards.

The network topology is illustrated in Figure 1, where User 1 can only receive time information from User 2, and Users 2 and 3 can synchronize time with each other equivalently. Solver Configuration: MATLAB ode45 solver with relative tolerance 10 ⁻³ selected for stiff dynamics in $f (x_{i}, t)$ until the users are synchronized.

Figure 1.

Synchronization topology model for three users.

The simulation runs for 10 s with the smallest sampling time set at 1 millisecond, meaning each iteration occurs every 1 millisecond. The state equations have specific parameter settings as follows:

\begin{array}{l} A = [\begin{array}{l} 0 & 1 & 0 & 0 \\ - 3 & - 1 & 3 & 0 \\ 0 & 0 & 0 & 3 \\ 2 & 0 & - 2 & 0 \end{array}], \\ B = {[\begin{array}{l} 0 & 1 & 0 & 0 \end{array}]}^{T}, \\ f (x_{i}, t) = {[\begin{array}{l} 0 & \begin{array}{l} 0 & 0 \end{array} & - 0.45 (\sin x_{i}) \end{array}]}^{T} . \end{array}

(42)

The state variable $f$ of each user device is affected by a sine function of state variable, with the corresponding Lipschitz constant is −0.45. The threshold function is selected as:

h (t) = \frac{1}{D^{i n}} (0.001 + 5 e^{- t}) A = [\begin{array}{l} 0 & 1 & 0 & 0 \\ - 3 & - 1 & 3 & 0 \\ 0 & 0 & 0 & 3 \\ 2 & 0 & - 2 & 0 \end{array}] .

(43)

Using MATLAB’s CVX toolbox, the calculations yield:

c = 3399.25, K = [\begin{array}{l} - 0.0079 & - 0.0008 & 0.0050 & - 0.0070 \end{array}] .

(44)

Figure 2 shows the results of the distributed synchronization scheme with an event-triggered mechanism. The figure illustrates that all four states of the user devices asymptotically synchronize. This means that the clocks of the users gradually achieve consistency and follow a common trajectory. Additionally, the entire synchronization process completes within 5 s, making it suitable for rapid deployment and adjustment in distributed power networks.

Figure 2.

Synchronization trend curve for three users.

To verify the convergence of the distributed synchronization algorithm in a network consisting of six users, consider the directed graph topology as illustrated in Figure 3. In this configuration, four key nodes 2, 3, 5, and 6 at the center of the network form a peer-to-peer synchronization ring. User 3 synchronizes with User 4 unidirectionally, User 4 synchronizes with User 5 unidirectionally, User 1 synchronizes with User 2 unidirectionally, and User 6 synchronizes with User 1 unidirectionally. The parameters for the system’s nonlinear dynamic equations are specified as equation (45).

\begin{array}{l} A = [\begin{array}{l} 0 & 1 & 0 & 0 \\ - 2 & - 1 & 2 & 0 \\ 0 & 0 & 0 & 2 \\ 2 & 0 & - 2 & 0 \end{array}], \\ B = {[\begin{array}{l} 0 & 1 & 0 & 0 \end{array}]}^{T}, \\ f (x_{i}, t) = {[\begin{array}{l} 0 & \begin{array}{l} 0 & 0 \end{array} & - 0.33 \sin (x_{i 3}) \end{array}]}^{T} \end{array}

(45)

Figure 3.

Synchronization topology model for six users.

The threshold function is selected as:

h (t) = \frac{1}{D^{i n}} (0.001 + 5 e^{- t})

(46)

Using MATLAB’s CVX toolbox, the calculations yield:

\begin{array}{l} c = 2251.48, \\ K = [\begin{array}{l} - 0.04703 & - 0.01336 & 0.02849 & - 0.02235 \end{array}] . \end{array}

(47)

As observed from Figure 4, even when the number of users is doubled and the network links become more complex, the adaptive adjustment scheme still achieves asymptotic synchronization of the user devices’ states within 5 s. The users’ clock states progressively converge and evolve along a common trajectory. This verification indicates that, as the network expands, the proposed distributed synchronization scheme remains effective, demonstrating strong robustness of the algorithm.

Figure 4.

Synchronization trend curve for six users.

Comparison of synchronization performance

In this section, we compare the performance of the proposed algorithm with existing algorithms based on the six-user topology mentioned earlier. The comparison algorithm for continuous communication is a classic distributed clock synchronization scheme from reference 2, with its input represented by equation (48).

u_{i} = c K \sum_{j \in N_{i}^{i n}} a_{i j} (x_{i} - x_{j}) = c K \sum_{j = 1}^{N} a_{i j} (x_{i} - x_{j})

(48)

For any user ii in a distributed network, appropriate choices of c and K can solve the synchronization problem of distributed devices. The provided protocol relies on continuous communication between agents, requiring continuous information exchange between nodes to update inputs. Since it is not event-triggered, the limits of this algorithm can serve as an upper bound for the synchronization error of event-triggered schemes, validating whether the synchronization error can asymptotically converge to this upper bound to assess the effectiveness of the distributed synchronization algorithm proposed in this chapter. This algorithm features continuous communication, but the bandwidth of power networks is limited, which may not meet this requirement. Another comparative algorithm is an event-triggered scheme with no node priority, designed based on the event-triggered function from reference 1.

Firstly, the convergence of the event-triggered scheme without node priority is validated. Simulations are conducted in a network scenario with six users, keeping the topology and parameter settings unchanged, and the threshold function is selected as:

h^{'} (t) = (0.001 + 5 e^{- t})

(49)

As Figure 5 shows, the comparison of synchronization errors for three different communication mechanisms—continuous communication, event-triggered synchronization without node priority, and event-triggered synchronization with node priority is analyzed. It is calculated that the upper bound for synchronization error is 0.2139. By adjusting the lower bound of the threshold function, synchronization error can be made arbitrarily small. Therefore, reducing the triggering threshold function will bring the synchronization error closer to zero. Although the frequency of information exchange between user nodes decreases, the synchronization error approaches that of the comparison algorithm without priority consideration. In this case, the synchronization precision of the node priority scheme is approximately 2.106 ms. Further adjustment of the control parameters can meet the high-precision synchronization requirements of new power applications.

Figure 5.

Comparison of synchronization error.

The control parameter calculation algorithm are shown in Table 2. In Table 3, analysis reveals that by reducing the number of synchronization instances at the central node, the pressure on concurrent node interactions is decreased, and system clock synchronization can still be achieved, which aligns with the experimental expectations. However, due to significant delay fluctuations of user 1, the synchronization error of user 2 in the one-way time transfer increases, resulting in a slight increase in synchronization counts. Aggregation nodes 3, 5, and 6 have shown noticeable improvements, with the average update frequency of the nodes decreasing by approximately 28% (Table 3).

Table 2.

Control parameters calculation algorithm.

Algorithm 2: Control parameter calculation algorithm
1: function compute K
2: Solve the linear matrix inequality for the variable $P = P^{T}, P ≻ 0, τ > 0, μ > 0$ :

3: Obtain $K = - B^{T} P^{- 1}$
4: end
5: function compute c
6: Calculate connectivity $a (L) = \min_{x \neq 0, x \cdot r = 0} \frac{x^{T} (RL + L^{T} R) x}{2 x^{T} Rx},$
7: Select coupling gain $c \geq τ / a (L)$
8: end

Table 3.

Clock synchronization frequency comparison for six users.

	User1	User2	User3	User4	User5	User6	Average node update frequency
Without improved threshold	70	30	20	43	40	10	67.5
Improved threshold	93	35	7	56	22	5	48.75

Conclusion

This paper proposes a distributed clock synchronization architecture based on an event-triggered mechanism and node priority optimization, which replaces centralized public network timing with direct node-to-node information exchange. This approach establishes a unified time standard while reducing the synchronization burden on critical nodes like CCOs and power grids. User devices achieve millisecond-level precision by adjusting clocks via one-hop neighbor data, with MATLAB simulations validating convergence in three-user and six-user networks—synchronization errors approach the lower bound of continuous communication, and node priority reduces update frequency by 28% for critical nodes, minimizing information exchange. The scheme reduces the information exchange burden on nodes, while ensuring high-precision clock synchronization for local user devices in new power applications.

Footnotes

ORCID iDs

Yitao Zhao

Zhiyu Xia

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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