**1. Introduction**

With the emergence of new phenomena, such as the widespread interconnection of power grids and the high penetration of renewable energy, in the past decades, cascading failures of power systems have caused several large-scale power outages worldwide, such as the power outages in California [1], and the disconnection accident in the power grid of Europe [2], which have caused huge economic losses and threatened the stable operation of power grids.

Due to its good technical and economic benefits in large capacity, long-distance, and flexible transmission, DC transmission technology has been widely used in long-distance power transmission, power grid interconnection, and other aspects. DC transmission technology has improved the ability of friendly large-scale renewable energy and effectively solved the imbalance between regional supply and demand of electricity [3]. With the continuous development of HVDC transmission technology and the application of many DC projects, China has built a large-scale complex AC/DC hybrid power grid [4,5]. However, the DC blocking fault in the DC transmission converter station will inevitably transfer the power from the DC transmission line to the AC transmission line, which will cause the power flow of the AC transmission line to exceed the thermal stability limit power and the

**Citation:** Tu, Y.; Jiang, L.; Zhou, B.; Sun, X.; Zheng, T.; Xu, Y.; Mei, S. Optimal Configuration of Battery Energy Storage for AC/DC Hybrid System Based on Improved Power Flow Exceeding Risk Index. *Electronics* **2023**, *12*, 3169. https:// doi.org/10.3390/electronics12143169

Academic Editors: Luis Hernández-Callejo, Jesús Armando Aguilar Jiménez and Carlos Meza Benavides

Received: 20 June 2023 Revised: 15 July 2023 Accepted: 18 July 2023 Published: 21 July 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

cascading failure. And then blackout accidents of power systems will happen, which seriously influence the local society and economy [6,7]. Therefore, to ensure the safe and stable operation of the AC/DC hybrid system, it is of great practical significance to eliminate the out-of-limit power flow caused by DC power transfer in AC transmission lines.

The energy storage has good dynamic active and reactive power regulation capabilities, and it can adapt to operational control requirements of different time scales. To reduce the load shedding after the system failure, improve the system operation flexibility and stability, and ensures the safe, reliable, and efficient operation of the AC/DC hybrid system, the DC power transfer of the AC/DC hybrid system is eliminated by using the energy storage [8]. However, due to the different impact of DC power transfer on other lines, the energy storage near the fault location is used to eliminate the power transfer after the fault disturbance (DC bi-polar blocking) in the AC/DC hybrid system, some sensitive and vulnerable transmission lines still have the problem of power flow exceeding the limit value. It is necessary to quickly identify the sensitive line set that has a great impact on the transmission power, and configure the energy storage in this node line, which can quickly eliminate the power limit, improve the system stability, and prevent the occurrence of major accidents.

At this stage, the method of identifying the vulnerability of the power system based on the dynamic characteristics of the power grid has been widely used [9]. In [10], the risk theory assessment method is used to identify key lines by simulating the hidden fault model in the chain fault, but it requires many simulation results to determine the probability of line disconnection through tests, which increases the workload and is difficult to achieve online application and reduces the practicality of the project. In [11], a vulnerability assessment method of power grid cascading fault propagation elements based on power flow entropy is proposed, which can distinguish the vulnerability of branches from impact and consequence. This method accurately models the physical characteristics of the power grid and can improve the simulation speed by reducing the fault search space. However, there is still a contradiction between sampling times and simulation accuracy, which is difficult to achieve online application. In [12], a comprehensive index is proposed to identify the vulnerable lines, which applies the impact vulnerability to represent its impactresistance ability and the transfer vulnerability to represent the damage caused by its removal from the system. However, this method does not consider the margin of power flow out-of-limit capacity of other lines after the disconnected transmission line. The method of identifying the sensitive and vulnerable transmission lines based on power flow exceeding risk index is proposed in [13]. The method would not have to repeatedly calculate the impedance matrix of the line disconnection and connection. But it is difficult to apply to AC/DC hybrid systems, weak power grids, and other power systems. The scope of the application is limited. The improved power flow exceeding risk index is used for AC/DC hybrid systems and other various power systems. And this method preserved the advantages of traditional methods.

The optimal configuration of BESS is mainly to determine its optimal access location and capacity, to better play its performance, and to improve the absorption rate of renewable energy. In [14,15], the proposed coordinated operational planning for wind farms with BESS is that it can reduce the impacts of wind power forecast errors. Considering the uncertainty and curtailment rate constraint of wind power, reference [16] focuses on the BESS configuration method in wind farms. In [17], the capacity allocation of BESS is used to smooth wind power fluctuations, and the BESS capacity size at different confidence levels is studied. In [18], this paper proposes a bi-level optimal energy storage system (ESS) siting and sizing algorithm to mitigate the voltage deviation in distribution networks. A capacity allocation method of BESS in secondary frequency regulation with the goal of maximum net benefit is proposed in [19]. The literature [14–19] focuses on a single application scenario, such as reducing prediction error, improving new energy consumption, and ensuring power grid stability to achieve BESS configuration, which has significant limitations. They do not fully explore their advantages in coordinated operation or multiple application

scenarios. Large BESS capacity needs to be configured, and the utilization rate of BESS is low.

References [20–22] propose an optimized configuration method for the coordinated operation of BESS and renewable energy. Collaborative configuration of distributed generation and BESS in microgrids considering the state of health is studied in [23]. In [24], considering the uncertainty of the net load, this study provides an approach to analyzing the BESS demand capacity for peak shaving and frequency regulation. In reference [25], the feasibility and compatibility of using such idle capacity and power of BESS to participate in the electricity energy market and reserve ancillary service market are explored, and a coordinated operation strategy for the three application scenarios of BESS is proposed to improve its utilization. When the power system is in a steady state, to achieve economic efficiency, ensure grid stability, and improve the utilization rate of BESS, the optimization configuration of BESS for the collaborative operation of BESS and renewable energy and multiple application scenarios of BESS services is studied by domestic and foreign scholars. But the frequent occurrence of extreme weather would seriously affect the safe and stable operation ability of the power grid. It is necessary to study the optimal configuration of BESS considering the influence of extreme weather on the power grid. This can enhance the safe and stable operation capacity of the power grid.

To cope with the impact of extreme weather, such as typhoons and freezing rain, on the power grid, BESS has been configured to improve the reliability and flexibility of the power grid in recent years. Literature [26] takes the load-shedding cost of the system under extreme events as the toughness index and studies the optimal allocation of BESS considering the toughness of the distribution network. In [27], a distributed energy storage planning model for the distribution network considering the influence of typhoon weather is established, and a decomposition collaborative solution method based on the Benders decomposition is proposed. In [28], a new quantitative index of toughness and formulates of a method of BESS planning were proposed to enhance the seismic capacity of the distribution network. Literature [29] proposed a distribution network BESS planning method considering toughness and established a two-stage robust optimization model, which can effectively ensure the uninterrupted power supply of important loads. The above literature configures BESS to improve the flexibility or toughness of the power grid by ensuring a continuous power supply of important loads in extreme weather. However, with the increase in the penetration rate of new energy, the probability of power grid failures has increased, such as exceeding the power limit of transmission lines and cascading faults in the power grid. It is very necessary to allocate BESS reasonably after a power grid failure, such as to quickly eliminate the over-limit of AC line power and improve the stability of the system; how to reasonably configure BESS after the DC locking fault occurs in the AC/DC hybrid system.

And then, a large amount of research has been conducted domestically and internationally on solution methods for BESS optimization configuration. Intelligent optimization algorithms, such as genetic algorithm [30] and particle swarm optimization (PSO) [31], have been widely applied. The optimization configuration method proposed provides a good reference for the solution in this article.

Aiming at the advantages and disadvantages of the existing research, considering the millisecond level active dynamic response capability of the BESS system, an optimized configuration of BESS in the AC/DC hybrid system based on the improved power flow exceeding risk index is proposed. Firstly, the improved power flow exceeding risk index is established to evaluate the sensitivity and vulnerability of other lines to the transferred power flow after the branch is disconnected. Secondly, the Dijkstra algorithm is used to find out the shortest path of the closed loop formed by the breaking line, and the key nodes are selected as the candidate sites for BESS by calculating the improved power flow exceeding risk index of the shortest path. Finally, a multi-objective function with the maximum sum of improved power flow exceeding risk index and the minimum annual investment cost of BESS is established, and particle swarm optimization (PSO) is used to obtain the optimal configuration scheme of BESS in the AC/DC hybrid system. Meanwhile, the BESS system adopted the active power control strategy, including plant-level control and local control, which quickly eliminates the power exceeding the limit of the AC line and suppresses the power fluctuation of the power grid.

This paper is organized as follows: The identification of sensitive and vulnerable lines is talked about in Section 2. The mathematical model for the optimal allocation of BESS is proposed in Section 3. The Model-solving method is given in Section 4. In Section 5, the effectiveness and feasibility performance of the proposed method are examined on the CEPRI36V7 grid model. Section 6 is the conclusion.

#### **2. Identification of Sensitive and Vulnerable Lines**

#### *2.1. Line Outage Distribution Factor*

If line A of the AC/DC hybrid system is faulty, and it causes the line disconnection (DC line causes blocking fault, etc.), which causes the transfer of active power flow in the system, that is, the active power flow of other lines is changed. The relationship between the change of normal line power flow and the original power flow of the disconnected line can be expressed by the Line Outage Distribution Factor (LODF) [32]:

$$D\_{R-A} = \frac{\Delta P\_{R-A}}{P\_A} \tag{1}$$

where Δ*PR*−*<sup>A</sup>* is the change of line *R*'s active power flow after line *A* is disconnected; *DR*−*<sup>A</sup>* is the LODF that causes the change of line *R*'s active power flow after line *A* is disconnected; *PA* is the steady-state initial active power of line *A*.

Assuming that the nodes at both ends of line *A* are *i* and *j*, and the injected active power remains unchanged before and after disconnection, the change of node active power flow caused by line *A* disconnection is

$$
\Delta P = \begin{bmatrix} 0 \cdots 1 \cdots \cdots -1 \cdots \cdot 0 \end{bmatrix}^T \\
 P\_A = M\_A P\_A \tag{2}
$$

where *MA* is the node-branch associated *n* × 1 order column vector of branch *A*, and the row corresponds to the node number.

The *n* × 1 order change Δ*θ* of node voltage phase angle caused by line *A* disconnection. Δ*θ* can be expressed as

$$
\Delta\theta = \left(B - M\_A \mathbf{x}\_A^{-1} M\_A^T\right)^{-1} M\_A P\_A \tag{3}
$$

where *B* is the *n* × *n* order admittance matrix; the admittance matrix is sparse type; *xA* is the reactance of line *A*.

Then, the change of active power flow of branch *R* (*R* = *A*) caused by line *A* disconnection is

$$
\Delta P\_{R-A} = \frac{M\_R^T \Delta \theta}{\varkappa\_R} = D\_{R-A} P\_A = \frac{M\_R^T (B - M\_A \ge\_A^{-1} M\_A^T)^{-1} M\_A P\_A}{\varkappa\_R} \tag{4}
$$

where *MR* is the node-branch associated *n* × 1 order column vector of branch *R*; *xR* is the 1 order reactance of line *R*.

Let *<sup>B</sup>*−<sup>1</sup> <sup>=</sup> *<sup>X</sup>*, after simplification, the expression of *DR*−*<sup>A</sup>* is:

$$D\_{R-A} = \frac{X\_{R-A}/\chi\_R}{1 - X\_{A-A}/\chi\_R} \tag{5}$$

Among them,

$$X\_{R-A} = M\_R^T X M\_A \tag{6}$$

$$X\_{A-A} = M\_A^T X M\_A \tag{7}$$

where *X* is the *n* × *n* order impedance matrix; *XA*−*A*, *XR*−*<sup>A</sup>* is the 1 order self-impedance and mutual impedance between nodes of port *R* and port *A*, respectively.

After the DC blocking fault occurs in the AC/DC hybrid system, the influence of DC power transfer on the AC line can be measured by calculating the LODF of the AC line.

#### *2.2. The Improved Power Flow Exceeding Risk Index*

The improved power flow exceeding risk index takes into account the impact of power transfer on other lines and the margin of the out-of-limit capacity of line power flow. After considering the margin of the out-of-limit capacity of line power flow, it is not necessary to consider the problem of reverse power flow of other lines caused by power flow transfer separately, which reduces unnecessary calculations, and can better reflect the sensitivity and vulnerability of other lines, and identify the sensitive and vulnerable lines in the system. Under the change of power flow of line R caused by the disconnection of line A, the margin of the out-of-limit capacity of the power flow of line R can be expressed as follows:

When *DR*−*<sup>A</sup>* < 0, this is true, the expression of line power flow out-of-limit capacity margin is as follows:

$$
\Delta P' = \begin{cases}
\end{cases} \tag{8}
$$

When *DR*−*<sup>A</sup>* > 0, this is true, the expression of line power flow out-of-limit capacity margin is as follows:

$$
\Delta P' = \begin{cases}
\end{cases} \tag{9}
$$

Combined with the LODF, the improved power flow exceeding risk index is given to evaluate the sensitivity and vulnerability of other lines to the transferred power flow after the branch break, as follows:

$$
\Phi\_{R-A} = \frac{\Delta P'}{D\_{R-A} \cdot P\_{\text{A,max}}} \tag{10}
$$

where *P*A,max is the thermal stability limit value of the breaking line.

After a DC blocking fault occurs in the AC/DC hybrid system, the sensitivity and vulnerability of each AC line can be effectively evaluated by calculating the improved power flow exceeding risk index of each AC line. The smallest the improved power flow exceeding risk index of each AC line, the lower its ability to receive the transferred power flow, and the higher the improved power flow exceeding risk index. This paper selects the sensitive vulnerability line when the absolute value of the improved power flow exceeding risk index is less than 0.5. Secondly, after the DC blocking fault occurs in the AC/DC hybrid system, the DC power flow is mainly transferred to the shortest path that forms a closed loop with the DC line, so the shortest path set of DC power transfer needs to be searched.

#### *2.3. The Shortest Path Search Based on the Dijkstra Algorithm*

Using the knowledge of graph theory, the AC/DC hybrid system is simplified and abstracted into a graph G (V, E), where V represents the bus set in the grid, E represents the line set between buses, and the line side weight value is the line reactance. Then, the shortest path algorithm related to graph theory is adopted.

The shortest path search algorithms in graph theory include the Dijkstra algorithm, Floyd algorithm, etc. The Dijkstra algorithm has small time complexity and is easy to expand; the Floyd algorithm has high time complexity and space complexity, which increases the calculation amount. Its advantage is that it can be used to search the shortest path of the line with negative weight. Because of the scalability of the Dijkstra algorithm and the fact that there are no branches with negative weights in the graph, and to meet the requirements of fast calculation, the shortest path search based on the Dijkstra algorithm is adopted.

After the DC blocking fault occurs in the AC/DC hybrid system, the shortest path forming a closed loop with the DC line can be searched by using the Dijkstra algorithm, and then the sensitive AC line can be identified by combining the improved power flow exceeding risk index, and the power input node of the sensitive, vulnerable line is used as the candidate location for configuring BESS.
