## 1. Introduction

With the development of modern communication, computing, network and control technologies, the applications of information technology continue to expand. The combination of information and energy technology has become an inevitable trend of the development for future power systems. This combination has also spawned a new concept: smart grid, which is able to utilize advanced information technology to improve energy management [

1,

2,

3,

4]. With smart grid technologies, we are able to control energy flows in power systems more efficiently and precisely.

Communication networks play a key role in the operation and management of smart grids [

5,

6]. In the monitoring system of a smart grid, a large amount of data needs to be processed and analyzed for the control and dispatch of power systems. The performance of the end-to-end communication of network has an important impact on the real-time capability and reliability of the monitoring system. The end-to-end performance of a communication network concerns both network transmission performance and computational performance. In addition, based on the analysis of smart grid wide-area monitoring cases in [

7,

8], the data synchronization performance is also one of the basic requirements in smart grid monitoring systems.

The monitoring system for a smart grid is a typical network computing system [

9]. For analysis, the network computing system can be regarded as a service system. Stochastic queuing theory, which is developed based on theories of Poisson process and Markov process, plays an important role in the performance analysis of network service systems such as telephone and telegraph networks [

10]. However, with the development of computer network systems, network structures and network applications have become more complicated and diverse. The traffic flow in communication networks exhibits properties different from those of Poisson and Markov processes. Therefore, the stochastic queuing theory would produce large deviations when applied to analyze the modern computer networks [

11]. On the other hand, the stochastic queuing theory can only provide limited performance metrics, such as the average waiting time and variance of the service system. It cannot be used to obtain the deterministic analysis of the system performance. In order to tackle more complicated analysis of modern network service systems, network calculus theory was proposed [

12,

13].

The network calculus theory can be applied to the performance analysis for service systems. It can be divided into two branches: deterministic network calculus theory and stochastic network calculus theory; see, e.g., [

12,

14]. The deterministic network calculus theory can be used to calculate the upper and lower bounds for different kinds of maximum performance of a service system. For example, the maximum delay, the maximum backlog, etc. The stochastic network calculus theory is able to provide the probability distributions for the performance boundaries of a service system, such as the distribution of the maximum delay and the distribution of the maximum backlog. Within the field of smart grids, there are also a number of related research outputs based on communication networks and power grids [

15,

16,

17]. In [

18,

19], network calculus theory is used to construct a reliability model for a power system which consists of conventional power generation devices, loads and renewable energy sources such as photovoltaic panels and wind power generators, achieving better utilization of renewable energy in smart grids. In [

20], network calculus theory is employed to calculate the performance of communication systems in home area networks. Targeting at the operational stability and security of power systems, a bounded model of communication delay is proposed based on network calculus theory in [

21].

Despite the great success in applications within smart grids, network calculus theory is not able to handle the communication system with synchronous computing requirements [

7]. Unfortunately, there are vast data synchronization requirements in smart grid applications. For a communication system which requires synchronous computing, the end-to-end delay of the system is not only related to the transmission delay, but also related to the difference of delay in different channels, which is not considered in network calculus theory. Thus, deterministic network calculus theory cannot be directly used to provide an estimation for the upper bound of the smart grid monitoring system’s end-to-end delay.

In this paper, based on network calculus theory and the analysis for delay of synchronous communication system, it is shown that the problem to calculate the upper bound for end-to-end delay in the synchronous communication system can be transformed to a general maximum/shortest path selection problem in graph theory. Next, we propose a new method to provide an estimation for the upper bound of the smart grid monitoring system’s end-to-end delay. To show the feasibility and effectiveness of our proposed method, several numerical simulations are performed.

The importance and contributions of this paper are stated as follows:

- (1)
A class of typical communication model for the monitoring system is investigated in this paper. The transmission network in the smart grid monitoring system is modeled as a transmission service node, such that network calculus theory can be applied. In this sense, the analysis methods proposed in this paper can be used under most scenarios of monitoring systems in the field of smart grid.

- (2)
It is notable that due to the synchronous property of the smart grid applications, the original network calculus theory cannot be directly applied in the delay analysis discussed in this paper. Based on the network calculus theory, an upper bound for the end-to-end delay in the synchronous communication system is derived. The simulations demonstrate the feasibility of the proposed method. With the development of the smart grid systems, there will be more applications based on the monitoring systems, and the theoretical results obtained in this paper can be utilized to improve the reliability and efficiency of the smart grid systems.

- (3)
In this paper, three theorems are proposed as our main results. In Theorem 1, the upper bound for the transmission delay in a transmission service node with strict service curve is derived. The data transmission delay in different time periods are discussed in detail. In Theorem 2, the formula for upper bound of system’s delay with multiple times of data exchange is derived. In Theorem 3, a general upper bound for transmission delay in the considered system is proposed.

The rest of the paper is organized as follows:

Section 2 introduces the considered synchronous calculation model.

Section 3 presents the theoretical analysis for the upper bound of the end-to-end communication delay in the considered model.

Section 4 provides several numerical simulations to evaluate our main results. Finally, a conclusion is given in

Section 5.

## 2. Typical Models for the Synchronous Communication Systems

Consider the synchronous calculation and transmission model in

Figure 1. Such a model has been extensively studied in [

7], and it is a typical synchronous communication model for monitoring systems in smart grids.

In

Figure 1,

${R}_{1}\left(t\right)$ and

${R}_{2}\left(t\right)$ stand for the input of two sets of monitoring data, and the corresponding arrival curves of the monitoring data are

${\alpha}_{1}$ and

${\alpha}_{2}$. Similar to the data arrival curves proposed in network calculus theory [

22],

${S}_{1}$ and

${S}_{2}$ represent for the equivalent transmission service nodes where the monitoring data go through the control center, and the corresponding service curves are denoted as

${\beta}_{1}$ and

${\beta}_{2}$. Here, the data arrival curve is the characteristic curve which is used to describe the monitoring data. We denote

${S}_{3}$ as an equivalent computing service node, with the scaling function being

$S\left(n\right)$ and the calculation service curve being

$C$.

${S}_{4}$ is denoted as a follow-up service model, with its the service curve denoted as

${\beta}_{4}$. Before

${R}_{1}\left(t\right)$ and

${R}_{2}\left(t\right)$ enter into

${S}_{3}$, they go through a synchronous link which causes the equivalent transmission service curve changing. Let us denote

${\beta}_{1}^{\prime}$ and

${\beta}_{2}^{\prime}$ as the equivalent transmission service curves after the synchronization link. Here,

${\beta}_{1}^{\prime}$ and

${\beta}_{2}^{\prime}$ are functions of

${\beta}_{1}$ and

${\beta}_{2}$, i.e.,

${\beta}_{1}^{\prime}={F}_{1}\left({\beta}_{1},{\beta}_{2}\right),$ and

${\beta}_{2}^{\prime}={F}_{2}\left({\beta}_{1},{\beta}_{2}\right).$The smart grid’s wide area measurement system has three components: power monitor unit (PMU), communication network and controller. The operation parameters of the utility grid within different regions are measured by PMU. Based on the time scale from the global positioning system, such data is sent to the control center for analysis and procession. Let

${G}_{1}\left(t\right)$ and

${G}_{2}\left(t\right)$ be defined as the amount of data that is generated by Sensor

$1$ and Sensor

$2$ with time scale

$t$, respectively. Then, we have:

and:

Next, the flow ratio is defined. Here, we assume that the sensor has a synchronous clock, and the time scale of data is marked at the same time, i.e., if ${G}_{2}\left(t\right)\ne 0$, then ${G}_{1}\left(t\right)\ne 0$. We assume that ${G}_{2}\left(t\right)\ne 0$, and we define ${\rho}_{max}\triangleq max\frac{{G}_{1}\left(t\right)}{{G}_{2}\left(t\right)},$ and ${\rho}_{min}\triangleq min\frac{{G}_{1}\left(t\right)}{{G}_{2}\left(t\right)}.$ The data arriving at ${S}_{3}$ is synchronized. The aggregate received data calculation service curve is $C$. Then, we have the minimal computational service curve for ${R}_{1}$, which is $\frac{{\rho}_{min}}{1+{\rho}_{min}}C.$ Similarly, for ${R}_{2}$, the minimal computational service curve is $\frac{1}{1+{\rho}_{max}}C.$

According to the above assumptions and calculation model of unified service transmission, the equivalent end-to-end service model for monitoring data

${R}_{1}$ is obtained as follows:

where the notation

$\otimes $ stands for the convolutional operator. Similarly, the equivalent end-to-end service model for monitoring data

${R}_{2}$ can be obtained as follows:

In real-world scenarios, the monitoring data is normally the same. Hence,

${\rho}_{max}={\rho}_{min}=1.$ In the above model, both sensors

${R}_{1}$ and

${R}_{2}$ reach the service node

${S}_{3}$ simultaneously due to the synchronization process. Thus, their processing time is also the same. After the calculation processing, the data of sensor

${R}_{1}$ and

${R}_{2}$ go through the same service node

${S}_{4}$ with the same time delay. This refers to the so-called synchronization property; see, [

7], and the references therein.

Based on the synchronous property, the end-to-end delay of the synchronization system can be analyzed. We assume that the data of sensor

${R}_{1}$ always arrives earlier than that of

${R}_{2}$ during a given time period

$[0,t]$. This means that the data of

${R}_{1}$ always waits for that of

${R}_{2}$ in the section of synchronization. As soon as the data of

${R}_{2}$ arrives, it can be input into the service node

${S}_{3}$. Hence, regarding

${R}_{2}$ the equivalent service curve is not changed via the synchronized transmission link, and we have

${\beta}_{2}^{\prime}={F}_{2}\left({\beta}_{1},{\beta}_{2}\right)={\beta}_{2}.$ According to [

7], the end-to-end service curve of

${R}_{2}$ can be expressed as:

According to the service theorems in network calculus theory [

22] and the obtained public network flow model,

${\beta}_{2}$ and

${\beta}_{4}$ can be obtained directly. We have the upper bound of the end-to-end delay of

${R}_{2}$ being:

However, the equivalent transmission service curve of ${R}_{1}$ has been changed, since the data of ${R}_{1}$ cannot go through service node ${S}_{3}$ until the arrival of the ${R}_{2}$. Then, we have ${\beta}_{1}^{\prime}={F}_{1}\left({\beta}_{1},{\beta}_{2}\right)\ne {\beta}_{1}.$ In fact, due to the waiting time of data, the delay of ${R}_{1}$’s data may increase. Then, we have ${\beta}_{1}^{\prime}\le {\beta}_{1}.$ According to the synchronization property, the end-to-end delay of ${R}_{1}$ and ${R}_{2}$ are the same, even if ${\beta}_{1}^{\prime}$ is unable to be obtained. The upper bound of the end-to-end delay of ${R}_{1}$ can be expressed by (3).

The problem is that the conclusion of (3) was tenable, only if we assume that

${R}_{1}$’s data always arrives earlier than

${R}_{2}$’s. If

${R}_{1}$’s data arrives later than

${R}_{2}$’s after time

${t}^{\ast}$, the expression of the end-to-end service curve of

${R}_{1}$ is presented as (4), which shall be investigated in

Section 3:

Although the data of

${R}_{1}$ arrives later, most of the equivalent transmission curve of

${R}_{1}$ has already changed before

${t}^{\ast}$. So, the service curve of

${R}_{1}$ cannot be

${\beta}_{1}$, i.e., (4) cannot be the equivalent transmission service curve to

${R}_{1}$. In order to calculate the end-to-end delay in this case, the delay theory of the suspension service system is discussed in

Section 3.

## 3. Calculation of Equivalent Delay of Monitoring System

The main results of the equivalent delay calculation are provided as three theorems in this section.

#### 3.1. Delay Theorem of Suspension Service System

**Theorem** **1.** Consider an input$R\left(t\right)$through a service node which has the strict service curve $\beta \left(t\right)$. The system does not provide any service during ${t}_{1}<t<{t}_{2}$. We assume that ${R}^{\ast}({t}_{1})$ is known and time delay of the original system is $d\left(t\right)$. For $t>{t}_{1}$, the delay is denoted as ${d}^{\prime}\left(t\right)$. Then, ${d}^{\prime}\left(t\right)$ satisfies the following inequality:where ‘$\vee $’ refers taking the maximum value. Before time ${t}^{\ast}$, data of ${R}_{1}$ always arrives earlier than that of ${R}_{2}$. But after time ${t}^{\ast}$, on the contrary, ${R}_{2}$’s data arrives earlier. The service received by ${R}_{1}$ can be equivalent to a suspended service system, and the suspended time period is ${d}_{1}\left({t}^{\ast}\right)<t<{d}_{2}\left({t}^{\ast}\right)$, where ${d}_{1}\left({t}^{\ast}\right)$ and ${d}_{2}\left({t}^{\ast}\right)$ are obtained from (2) and (4), respectively. Because of the synchronization, the data which was supposed to be processed in ${d}_{1}\left({t}^{\ast}\right)$ by ${R}_{1}$ will not be completed until ${d}_{2}\left({t}^{\ast}\right)$, and this is equivalent to the system being suspended for ${R}_{1}$.

#### 3.2. Synchronization System Delay Analysis

Since the data sent by sensors is in accordance with a fixed sampling interval, we denote the time scale of such data as

${T}_{1}<{T}_{2}<{T}_{3}<\cdots <{T}_{n}$. Then, we have:

Let us introduce the following definitions.

**Definition** **1.** Let us denote ${\beta}_{R1}^{1}$, ${\beta}_{R1}^{2}$, ${\beta}_{R2}^{1}$, ${\beta}_{R2}^{2}$ as service curves of flow ${R}_{1}$ and flow ${R}_{2}$ before and after the synchronous link, respectively, then we have ${\beta}_{R1}^{1}={\beta}_{1},{\beta}_{R2}^{1}={\beta}_{2},$ and: **Definition** **2.** Denote ${d}_{i}^{x}\left(t\right)$ as the delay upper bound calculated by the equivalent service model using flow $i$, after the data arrival sequence changes for the $x$-th time. Denote ${D}_{i}\left(t\right)$ as the upper bound of the system delay before time $t$, when the data arrival order has already changed $i$ times. Then, we have: We assume that the flow arrival curves for ${R}_{1}$ and ${R}_{2}$ are ${\alpha}_{1}$ and ${\alpha}_{2}$, respectively, and data of ${R}_{1}$ always arrives earlier than that of ${R}_{2}$ at the synchronization service node no later than time ${T}_{{x}_{1}}$. The flow delay of ${R}_{2}$ is ${d}_{2}^{0}\left(t\right)$. If there is no synchronization mechanism, flow delay of ${R}_{1}$ is ${d}_{1}^{0}\left(t\right)$. Since ${R}_{1}$’s data always arrives earlier than ${R}_{2}$’s, the system delay should be ${R}_{2}$’s delay which is ${d}_{2}^{0}\left(t\right)$. According to network calculus theory, we have ${d}_{2}^{0}\left(t\right)=h\left({\alpha}_{2},{\beta}_{R2}\right).$

After time

${T}_{{x}_{1}}$,

${R}_{2}$’s data arrives earlier than

${R}_{1}$’s at the synchronization service node. Thereafter, the delay of

${R}_{1}$ should be taken as the system delay. However, due to the waiting time of flow

${R}_{1}$, the original system delay

${\beta}_{R1}$ has changed. According to the analysis in

Section 3.1, the equivalent time period of flow

${R}_{1}$ is obtained as follows:

Flow ${R}_{1}$ can be seen as the output of the suspensive service ${\beta}_{R1}^{2}$ which comes from service ${\beta}_{R1}^{1}$ first. The suspension time for service ${\beta}_{R1}^{2}$ is the length of the time period given in (6). Assume that the data from flow ${R}_{1}$ passes by ${\beta}_{R1}^{1}$, the output is ${R}_{1}^{1}\left(t\right)$ at time $t$, and the data from flow ${R}_{1}$ reaches the synchronization link at time $t\u2019$. Obviously, ${R}_{1}^{1}\left({t}^{\prime}\right)={R}_{1}\left(t\right).$

Since the considered system is a periodic sampling monitoring system, data can only be transmitted in a fixed time period. The delay of the data transmission needs to be taken into consideration only when data is transmitted. Let us define $t={T}_{m}.$. Thus, ${t}^{\prime}$ defined above stands for the time when data with time scale ${T}_{m}$ reaches the synchronize link.

Assuming that the output of flow

${R}_{1}$ by service

${\beta}_{R1}^{1}$, synchronization and service

${\beta}_{R1}^{2}$ at time

${T}_{m}$ is

${R}_{1}^{\ast}(t)$, then, for the suspended starting point

${T}_{{x}_{1}}+{d}_{1}^{0}\left({T}_{{x}_{1}}\right)$, we have:

Let us assume that after time scale

${T}_{{x}_{2}}$, data

${R}_{1}$ arrives the synchronization node before data

${R}_{2}$ with the same time scale.

${R}_{2}$’s traffic can be seen as the output of the suspensive service

${\beta}_{R2}^{1}$ which gets through service

${\beta}_{R2}^{2}$ first. The suspension time is:

Therefore,

${T}_{m}>{T}_{{x}_{2}}$, the upper bound of data delay be expressed as:

In accordance with the discussion of above, when

${T}_{\omega}\ge {T}_{{x}_{1}}$:

If

${d}_{1}^{1}\left({T}_{{x}_{2}}\right)={d}_{2}^{0}\left({T}_{{x}_{2}}\right)$, then the equivalent model of the service is not suspended, so

$\left({T}_{m}\right)={d}_{2}^{0}\left({T}_{m}\right).$ Otherwise, we can obtain:

The problem is that the time of

${x}_{1}$ and

${x}_{2}$ cannot be obtained with the existing theory in network calculus [

22]. The maximum of:

is the bound of the system, which is:

So, for

${T}_{{x}_{2}}\ge {T}_{\omega}\ge {T}_{{x}_{1}}$,

Furthermore, for

${T}_{m}>{T}_{{x}_{2}}$:

Given

${\theta}_{1}={\theta}_{2}$, according to (A1), we have:

So, for

${T}_{m}\ge {T}_{{x}_{2}},$ we have:

For

${T}_{m}>{T}_{{x}_{2}}$, the upper bound of the system delay is

${d}_{1}^{2}\left({T}_{m}\right)$. For

${T}_{{x}_{2}}\ge {T}_{\omega}\ge {T}_{{x}_{1}}$, the upper bound of the system delay is

${d}_{2}^{1}\left({T}_{\omega}\right)$. For

${T}_{{x}_{1}}\ge {T}_{\phi}$, the upper bound of the system delay is

${d}_{1}^{0}\left({T}_{\phi}\right)$. Then, we have:

And:

If

${\theta}_{1}={\theta}_{2}$, according to formula (A1), we have:

#### 3.3. Synchronization System Delay Upper Bound Theorem

**Theorem** **2.** The system’s delay upper bound of the $n$-th exchange of data arrival sequence before time ${T}_{m}$ can be expressed as:where${d}_{1}^{n}\left({T}_{m}\right)$and${d}_{2}^{n}\left({T}_{m}\right)$have different expressions, which are determined by the property of $n$. If $n$ is an odd number, and if the equivalent model of data stream 1 is used after the last change, then data stream 1 arrives sooner than data stream 2 before the first exchange. If the equivalent model of data stream 2 is used after the last change, then data stream 2 arrives sooner than data stream 1 before the first exchange. Therefore, we have: If n is an even number, and if the equivalent model of data stream 1 is used after the last change, then data stream 2 arrives sooner than data stream 1 before the first exchange. If the equivalent model of data stream 2 is used after the last change, then data stream 1 arrives sooner than data stream 2 before the first exchange. Therefore, we have: In the above theorem, the system’s delay upper bound of the $n$-th exchange of data arrival sequence before time ${T}_{m}$ is obtained. Next, for any given time $t$, we drive the expression of system’s delay upper bound.

Let us define $h\left(\alpha ,\beta ,t\right)=su{p}_{0\le s\le t}\left\{in{f}_{\tau \ge 0}\left\{\alpha \left(s\right)\le \beta \left(s+\tau \right)\right\}\right\}$.

**Theorem** **3.** At any given time t, the system’s delay upper bound can be expressed as $d\left(t\right)\le h\left(\alpha ,\beta ,t\right)$.

According to Theorem 3, we have

$h\left(\alpha ,\beta ,{t}_{1}\right)\le h\left(\alpha ,\beta ,{t}_{2}\right),{t}_{1}\le {t}_{2}$. According to Theorem 3,

${D}_{n}\left(t\right)$ in Theorem 2 can be written as follows (

$n$ is an odd number):

#### 3.4. The Method of Calculation of the Upper Bound Equivalent Synchronization System Delay

Furthermore, consider the case of

$p$-channel data, if it needs to be synchronized. The flows are represented as

$1,2,\cdots ,p$. According to the similar analysis in

Section 3.2 and

Section 3.3, the upper bound of the delays can be represented using the model as follows:

Therefore:

where

$f\left(t\right)$ can be equivalent to any path from node 0 to node

dest in

Figure 2.

$Maxf\left(t\right)$ can be equivalent to obtaining the maximum path.

There are

$pm+2$ nodes in this graph, and the nodes in the same row do not connect with each other. There is no connection between the nodes of the same column, but there is a connection between the two nodes in different rows and different columns. Besides, node 0 and any other node are connected. Node

dest is connected to the node km only, and the distance is 0. The defined distance is represented as

$l\left(nod{e}_{1},nod{e}_{2}\right)$. So, we have:

In

Figure 2, all paths from node 0 to node

dest constitute the value of

$f\left(t\right)$. For example, taking

${\theta}_{1}=1$,

${\theta}_{2}=2$,

${\theta}_{3}={\theta}_{4}=\cdots ={\theta}_{n}={\theta}_{2}$,

${x}_{1}=1$,

${x}_{2}=3$,

${x}_{3}=m$, we have:

It can be expressed as the path distance which is $l\left(0,1\right)+l\left(1,2m+2\right)+l\left(2m+2,pm\right)$.