Intelligence & Robotics

Open Access Research Article

Correspondence to: Prof. Chen Peng, School of Mechatronic Engineering and Automation, Shanghai University, No. 99, Shangda Road, Baoshan District, Shanghai 200444, China. E-mail: c.peng@shu.edu.cn

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© The Author(s) 2022. **Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, sharing, adaptation, distribution and reproduction in any medium or format, for any purpose, even commercially, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

This paper investigates the scheduling process for multi-area interconnected power systems under the shared but band-limited network and decentralized load frequency controllers. To cope with sub-area information and avoid node collision of large-scale power systems, round-robin and try-once-discard scheduling are used to schedule sampling data among different sub-grids. Different from existing decentralized load frequency control methods, this paper studies multi-packet transmission schemes and introduces scheduling protocols to deal with the multi-node collision. Considering the scheduling process and decentralized load frequency controllers, an impulsive power system closed-loop model is well established. Furthermore, sufficient stabilization criteria are derived to obtain decentralized ℋ_{∞} output feedback controller gains and scheduling protocol parameters. Under the designed decentralized output feedback controllers, the prescribed system performances are achieved. Finally, a three-area power system example is used to verify the effectiveness of the proposed scheduling method.

Multi-area power systems, round-robin scheduling, try-once-discard scheduling, load frequency control

With the vigorous development of power markets, the degree of interconnection among power grid regions is increasing. At present, power grids have developed into multi-area interconnected power systems^{[1, 2]}. Generally, there are two communication schemes to transmit data between neighborhood areas, that is, the dedicated channel and the open infrastructure. Since the latter has outstanding advantages over the former, such as higher flexibility and lower cost, it has been widely used in the communication of multi-area interconnected power systems^{[3-5]}.

An important indicator to measure power system operation quality is the fluctuation in frequency, and load frequency control (LFC) is widely used to maintain frequency interchanges at scheduled values and stifle frequency fluctuation caused by load disturbances^{[2, 6, 7]}. Under the open communication infrastructure, the sensor in each area collects data information. Then, data is transmitted to the decentralized controller under the shared but band-limited network. The corresponding control commands are issued to actuators in each area respectively. However, due to the introduction of the network, the design and operation of LFC face some new challenges, such as node collision, data loss and network-induced delay^{[8-10]}.

Due to different locations of sub-region power grids and the wide distribution of system components, multi-channel transmission is inevitable in multi-area interconnected power systems, while most of the existing results^{[10-13]} take the general assumption that sampled data is packaged into a single packet to transmit, which is not applied in large-scale multi-area interconnected power systems. On the other hand, the shared network has limited bandwidth, where the simultaneous transmission under multiple channels may cause node congestion. To solve this problem, scheduling protocols have been presented to decide which node to gain access to the communication network^{[14, 15]}.

Generally, multi-channel scheduling includes Round-Robin (RR) scheduling^{[16, 17]}, try-once-discard (TOD) scheduling^{[18, 19]}, and stochastic scheduling^{[20]}. Under the RR scheduling protocol, each node is transmitted periodically with a fixed period whose value is the total number of transmission channels. Under the TOD scheduling, the sensor node with the largest scheduling error has access to the channel. Recently, a time-delay analysis method has been discussed to derive stability criteria for networked control systems (NCSs) which are scheduled by the above three communication protocols^{[14, 21]}. Besides, a hybrid system method has been employed to analyze NCSs with variable delays under the TOD communication protocol^{[19]}, where a partial exponential stabilization criterion has been derived. The ^{[22]}, in which TOD protocol is used to schedule sampled data.

This paper studies the

(1) RR and TOD protocols are used to deal with the multi-node collision of large-scale power systems, which improve communication efficiency greatly. Compared with the existing LFC methods^{[10, 23, 24]}, the scheduling process under multi-area transmission schemes is investigated.

(2) An networked power system impulsive closed-loop model is well constructed, which covers the multi-channel scheduling, packet dropout, disturbance, and network-induced delays in a unified framework. Compared with the system without disturbance^{[19]}, this paper studies the anti-disturbance performance of the studied system. An

Throughout this paper,

Considering single-packet size constraints, the dynamic model of multi-area interconnected power systems with decentralized load frequency controllers is constructed in this section. An impulsive system model under TOD or RR protocol is established.

Diagram of the multi-area decentralized LFC under scheduling protocols is shown in Figure 1, where data transmission from sensors to decentralize controllers is scheduled by RR or TOD scheduling protocol.

The system model is represented as^{[23, 25]}:

where

The state and measured output signals are

where

The multi-area power system includes

In this paper, we take imperfect network conditions into account, such as data loss and network-induced delay. Denote the sequence after packet loss by

Let

Consider the scheduling error between output

where

Under RR scheduling protocol, the active node

In the following, we will design the decentralized LFC law under the communication network. Similar to^{[23]}, a PI controller is used in this paper:

Therefore, dynamic model of the scheduled power systems under the decentralized LFC Equation (8) and imperfect network environments can be formalized as

where

From Equation (5), one can obtain that

Define an artificial delay

From Equation (5) and Equation (9), the impulsive power system model can be

For system Equation (12), the initial condition of

Using scheduling scheme Equation (6) and Equation (7), this paper is to design the decentralized controller Equation (8) such that system Equation (12) is exponentially stable with a prescribed

In the following, we first derive sufficient criteria under scheduling scheme Equation (6) and Equation (7) to ensure the exponential stability of system Equation (12) with a prescribed

Construct Lyapunov-Krasovskii functional candidate:

where

**Remark 1***is introduced to cope with reset conditions, which is continuous on *

**Theorem 1***Under TOD scheduling scheme Equation (6), for given scalars *

*are feasible for *

**Proof**: Differentiating ^{[26]}, we can obtain

where

We use the reciprocally convex approach^{[27]} to deal with the cross item in Equation (18). By using Schur's complement, one can get

In the following, we will prove that

(ⅰ) with

Since

It follows from

where

Then taking Equation (17) and Equation (22) into account, denoting

Hence, one can conclude that

From Equation (24), we have

which implies that

It follows from

For

Clearly, there exists the maximum of

Therefore, the exponential stability of the system Equation (12) with

Next, we will show that

(ⅱ) under the zero initial condition, the inequality

From Equation (19), we obtain that for

Integrating Equation (27) on

Then, by summing Equation (28) on

Under the zero initial condition,

**Remark 2***The feasibility of the linear martix inequalities has been sufficiently explained in ^{[23, 28]}. Due to page limitations, this part is omitted here*.

Construct the following Lyapunov-Krasovskii functional candidate:

where

Similar to Theorem 5.2 in^{[29]}, we establish the following result.

**Theorem 2***Under RR scheduling scheme Equation (7), given *

**Proof**: The detailed derivation process can refer to^{[29]} and the proof of Theorem 1, which is omitted here due to the limited pages.

**Theorem 3.3.***Under TOD scheduling scheme Equation (6), for given matrices *

*is solvable, and the controller gain is given by *

**Proof**: Let

and their transposes, respectively. For the nonlinear terms ^{[27]}, one can obtain

Similar to Theorem 5.2 in^{[29]}, we establish Theorem 4.

**Theorem 4***Under the RR scheduling scheme Equation (7), for given matrices *

**Proof**: The detailed derivation process can refer to^{[29]} and the proof of Theorem 3, which is omitted here.

In the following, we use a three-area power system^{[23, 24, 28]} interconnected by the shared communication network to demonstrate the effectiveness of main results. Parameters are listed in Table 1.

Table 1

Configuration of three-area power systems

0.30 | 0.37 | 0.05 | 1.0 | 10 | ||

0.17 | 0.40 | 0.05 | 1.5 | 10 | ||

0.20 | 0.35 | 0.05 | 1.8 | 12 | ||

**Case 1:** Stability of the studied system under TOD scheduling scheme Equation (6) and

Assume parameters are chosen as

Correspondingly, state responses and the switching behavior of active nodes are illustrated by Figure 2 and 3, respectively. Clearly, the designed dynamics output feedback controllers can stabilize the three-area power system under the TOD scheduling scheme Equation (6) in the absence of disturbance.

**Case 2:** Stability of the studied system under RR scheduling scheme Equation (7) and

To compare with the TOD protocol Equation (6), we use the same parameters as case 1. We apply Theorem 4 which yields

**Case 3:** Stability of the studied system under TOD scheduling scheme Equation (6) and

According to Theorem 1-4, TOD protocol can degrade into RR protocol under certain conditions. In this case, we take the TOD protocol Equation (6) as an example to verify the anti-disturbance performance of the studied system. Under the disturbance

This paper has designed

All authors contributed equally.

This work was supported in part by the National Natural Science Foundation of China under Grant 62173218, and the International Corporation Project of Shanghai Science and Technology Commission under Grant 21190780300.

Not applicable.

All authors declared that there are no conflicts of interest.

Not applicable.

Not applicable.

© The Author(s) 2022.

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