A Method for Power Flow Calculation in AC/DC Hybrid Distribution Networks Considering the Electric Energy Routers Based on an Alternating Iterative Approach

Abstract

With the advancement of new power system construction, distribution networks are gradually transforming from being a simple energy receiver and distributor to being an integrated power network that integrates sources, networks, loads, and energy storage with interactive and flexible coupling with the upper-level power grid. However, traditional distribution networks lack active control and distribution capabilities, failing to meet the demands of network transformation and upgrading. To address this issue, this paper proposes a method for solving AC/DC power flow calculation considering an electric energy router (EER) based on an alternating iterative method. Initially, the model for the multi-port EER and three types of power flow models for the AC distribution network, DC distribution network, and EER are constructed. By leveraging the properties of the EER and the hybrid power flow calculation model, a method is proposed for calculating the power flow in the AC/DC hybrid distribution network considering the EER. Finally, by solving the power flow in a medium- and low-voltage AC/DC distribution system, the adaptability of the proposed method is compared. The results demonstrate that the AC/DC hybrid distribution network power flow calculation method established in this paper, which incorporates the EER, possesses high accuracy and adaptability, with an error margin of less than 0.05%.

1. Introduction

Distribution networks, as the final link in the power grid directly connected to end-users, constitute an essential component of the emerging power system paradigm. With the advancement of new power systems, distribution networks are gradually transforming from mere conduits for electricity delivery into integrated platforms that seamlessly incorporate generation sources, grids, loads, and energy storage while flexibly interfacing with upper-level power grids. Their pivotal role in facilitating the local consumption of distributed power sources and accommodating novel load demands is increasingly pronounced [1].

Traditional distribution networks primarily rely on centralized control to regulate power and frequency through generators [2], but they lack active control and distribution capabilities, which fail to meet the requirements for the transformation and upgrading of distribution networks. Therefore, there is a need for new types of distribution equipment to promote the development of the energy internet. In this context, the concept of the energy internet architecture has emerged [3,4]. Electric energy routers (EERs), as a pivotal component in the energy internet [5], integrate information flow and energy flow, enabling the efficient transmission of distributed renewable energy sources [6]. The authors of references [7,8] address the overload issues that arise after the integration of distributed renewable energy. They propose a method for configuring power routers at the access points and capacities in the distribution network. This method leans towards graph theory analysis and the overall operation of the distribution network, but it does not study the configuration of the number of power routers and ports. The authors of references [9,10] study the characteristics of AC/DC hybrid distribution networks in terms of power transmission and coordinated control while fully considering the network’s structural features and operational requirements. Based on this, they design a power electronics configuration suitable for EERs and develop a corresponding analysis model. The authors of reference [11] combine the functions of EERs and electric energy exchangers to construct a new structure tailored for the energy internet architecture.

Currently, extensive research has been conducted on the topology and control strategies of EERs. However, there is a clear research gap in the area of power flow calculation and optimal configuration. Power flow calculation is an analytical method used by power systems to determine the operating state of the network, providing valuable insights for system planning and operation. At present, the hybrid connection of AC and DC has become a prominent research hotspot in the field of distribution networks [12]. Existing power flow calculation methods for AC/DC hybrid distribution networks mainly include alternating iteration methods [13] and unified iteration methods. The authors of reference [14], in response to the asymmetry of the network structure and the volatility of distributed new energy, proposed a suitable power flow calculation method for AC/DC hybrid distribution networks. The authors of references [15,16] analyzed converter power losses comprehensively, effectively transforming them into impedance and admittance parameters within the circuit model to build a steady-state model applicable to AC/DC hybrid distribution networks.

On this basis, this paper proposes a power flow calculation method for AC/DC systems considering the EER. Firstly, the model of the multi-port EER is constructed, and its operational losses and control characteristics are analyzed. Then, three power flow models are constructed using the Newton–Raphson method for the AC distribution network, DC distribution network, and EER. By incorporating the characteristics of the EER and utilizing the hybrid power flow calculation model, a power flow calculation method for AC/DC hybrid distribution networks considering the presence of the EER is proposed. Eventually, by solving the power flow in a medium- and low-voltage AC/DC distribution system, the adaptability of the proposed method is compared.

This paper establishes an AC/DC power flow calculation method that incorporates EERs, aiming to address the issues of energy concentration and redistribution with the large-scale integration of renewable energy. It also provides a new research direction for EERs in power flow calculation and optimization.

2. The Construction of a Multi-Port EER Model

2.1. Steady-State Model of EER Port

As indicated in reference [17], the EER operates by managing and distributing energy flow according to instructions from the information flow, utilizing flexible power electronics technology. The information flow transmits the real-time status of the power grid by uploading the state information of the energy flow to the dispatch center. The functions of the EER include AC/DC voltage conversion, multi-voltage level voltage isolation, energy aggregation, and AC/DC load power supply. The authors of references [18,19] propose the national standards and system architecture for the EER, as illustrated in Figure 1.

To simplify its system framework and corresponding functions, a four-port EER topology containing power electronic devices is depicted in Figure 2.

The distribution of the four ports in the EER is shown in Figure 2. Regarding AC ports, the medium-voltage AC port is derived from the Voltage Source Converter (VSC), while the low-voltage AC port is obtained from the Dual Active Bridge (DAB) converter [20]. Concerning DC ports, the medium-voltage DC port derives its power directly from the internal DC bus due to its identical voltage, while the low-voltage DC port originates from the DAB [21]. These aforementioned ports are typical and modularly independent, allowing for the flexible selection of port types and quantities according to actual engineering needs.

The input and output ports establish a connection between the EER and the external AC/DC distribution network [22], with each port efficiently channeling energy to the internal DC bus through a dedicated DC/DC converter. The EER features distinct types of ports: AC ports for alternating current connections and DC/DC ports for direct current connections. The following will provide a comprehensive analysis and modeling of the two distinct types of ports separately.

(1) AC/DC Converter Port Modeling.

When modeling the AC port of the EER, a modified Augmented Nodal Analysis (MANA) is used for the modeling process. An additional node is expanded to the right side of the port, which is connected to the AC grid, along with an added transmission line. The inclusion of this supplementary node and line effectively eliminates the need for complex voltage and phase angle conversion procedures, thereby simplifying the overall port model [23]. Combined with its power electronic architecture, Figure 3 illustrates the steady-state equivalent model of the EER.

As shown in Figure 3, node i is connected to the AC port of the EER, with a voltage of $U_i\angle {\theta }_i$ [24]. The newly added node and line are node k and line ik, with an AC voltage of $U_k\angle {\theta }_k$. The admittance between node i and node k is $Y_{ik}=G_{ik}+j{B}_{ik}$, and the port capacitance is $B_{si}$. In accordance with Kirchhoff’s current and voltage laws, the active power $P_i^{ac}$, reactive power $Q_i^{ac}$, and parameters of the AC/DC converter ${\theta }_{ik}$ injected into the AC port of the EER are presented as follows [18]:

$$\{\begin{array}{l}\begin{array}{l}{P}_i^{ac}=\left(G_{ik}\cos {\theta }_{ik}+B_{ik}\sin {\theta }_{ik}\right)U_i{U}_k\hfill \\ Q_i^{ac}=-\left(B_{ik}\cos {\theta }_{ik}-G_{ik}\sin {\theta }_{ik}\right)U_i{U}_k\hfill \end{array}\\ {\theta }_{ik}={\theta }_i-{\theta }_k\end{array}$$

(1)

In the equations, $U_i$, $G_{ik}$, $B_{ik}$, and $B_{si}$ are typically determined by the power system and remain constant, while the EER can control the magnitude and direction of $P_i^{ac}$ and $Q_i^{ac}$ by altering the converter’s ${\theta }_{ik}$ and $U_k$.

Considering the equivalence of losses between the AC/DC converter and line ik using the extended node method, it can be inferred that the power $P_k^{ac}$ flowing into the AC side is equal in magnitude to the power $P_{er}^{dc}$ flowing into the DC side. The corresponding formula is shown as follows:

$$\{\begin{array}{l}{P}_k^{ac}=\left(G_{ik}\cos {\theta }_{ik}-B_{ik}\sin {\theta }_{ik}\right)U_i{U}_k\\ P_k^{ac}=P_{er}^{dc}=U_{er}^{dc}{I}_{er}^{dc}\end{array}$$

(2)

In the given formula, $U_{er}^{dc}$ is indicative of the voltage found on the DC side, $I_{er}^{dc}$ denotes the current present on the DC side, and the power, denoted by $P_k^{ac}$, that flows into the DC side is calculated by the multiplication of the DC voltage and current.

The relationship between the voltages $U_k$ and $U_{er}^{dc}$ on either side of the converter is shown as follows:

$$U_k=m{U}_{er}^{dc}$$

(3)

In the formula, m is the equivalent modulation coefficient, with a value range of 0 to 1.

(2) DC/DC Converter Port Modeling

When modeling the DC/DC converter port of the EER, a resistor can be used to simulate the losses incurred by the converter [25]. The DC/DC converter port model is shown in Figure 4 [26]. Here, n is the conversion ratio of the converter, $R_k$ represents internal losses, and $U_{DC}$ and $P_{DC}$ refer to the voltage and power on the primary side of the converter, respectively. Meanwhile, $U_{dc}$ and $P_{dc}$ correspond to the voltage and power on the secondary side of the converter, respectively [27]. Additionally, $I_{DC}$ and $I_{dc}$ signify the currents on both sides of the converter.

As depicted in Figure 4, the voltage relationship between the primary and secondary aspects of the converter can be formulated as follows:

$$U_{DC}=R_k{I}_{DC}+n{U}_{dc}$$

(4)

The current and power relationships on both sides of the converter can be represented as follows:

$$\{\begin{array}{l}{I}_{DC}={\displaystyle \frac{n^2{U}_{DC}}{R_k}}-{\displaystyle \frac{n{U}_{dc}}{R_k}};P_{DC}={\displaystyle \frac{n^2{U}_{DC}^2}{R_k}}-{\displaystyle \frac{U_{DC}{U}_{dc}}{R_k}}\\ I_{dc}={\displaystyle \frac{n{U}_{DC}}{R_k}}-{\displaystyle \frac{U_{dc}}{R_k}};P_{con}={\displaystyle \frac{n{U}_{DC}{U}_{dc}}{R_k}}-{\displaystyle \frac{U_{dc}^2}{R_k}}\end{array}$$

(5)

When the external DC grid and the internal DC voltage bus are at the same voltage level, and there are no strict power control requirements for the port, it is possible to eliminate the need for a DC/DC converter.

2.2. The Construction of the Internal Power Model of the EER

The external interfaces provided by the EER are composed of (M + N) AC/DC converters and (K + T) DC/DC converters, with each port operating independently. The EER is connected to external AC and DC distribution systems through its input and output interfaces, where each connection port uses DC/DC converter technology to aggregate energy to the internal DC bus. Therefore, neglecting the losses of the isolation-level DC/DC converters, the two internal DC buses can be considered as a unified entity, and the internal power conservation equation can be obtained as follows:

$${\displaystyle \sum _{w\in M+N}{P}_{er,w}^{dc}}+{\displaystyle \sum _{w\in K+T}{P}_{con,w}}=0$$

(6)

In the formula, $P_{er,w}^{dc}$ represents the active power flowing into the internal DC bus from the $wth$ AC/DC converter port, and $P_{con,w}$ denotes the active power flowing into the internal DC bus from the $wth$ DC/DC converter port.

2.3. Loss Situation and Control Methods of EER

The losses of the multi-port EER primarily comprise VSC and DAB losses, with the loss distribution of each port detailed in

Table 1. Types of losses for each port.

Port Type	Number of Ports	VSC Losses	DAB Losses
Medium-Voltage AC Port	M	√	×
Low-Voltage AC Port	N	√	√
Medium-Voltage DC Port	K	×	×
Low-Voltage DC Port	T	×	√

.

(1) VSC Losses

As shown in the table above, for VSC losses, which are primarily due to the voltage conversion of AC/DC converters, the losses at the AC/DC converter ports are modeled as equivalent to the line ik losses using methods like the extended node method. Combining Formulas (1) and (2), the VSC loss equation $P_{loss,i}^{VSC}$ for node i is written as follows:

$$P_{loss,i}^{VSC}=P_k^{ac}-P_i^{ac}=2m{G}_{ih}\cos {\theta }_{ik}{U}_i{U}_{er}^{dc}$$

(7)

(2) DAB Losses

As shown in the table above, for DAB losses, the DAB modules are present only at the low-voltage AC and low-voltage DC ports. This paper approximates the loss of the DAB device as 2% of the power flowing through it. Combining Formulas (1)–(3) and Formula (5), the DAB losses ($P_{loss,i}^{DAB,ac}$) at the low-voltage AC port and ($P_{loss,i}^{DAB,dc}$) at the low-voltage DC port for node i are as follows:

$$\{\begin{array}{l}{P}_{loss,i}^{DAB,ac}=0.02P_{er}^{dc}=0.02m\left(G_{ik}\cos {\theta }_{ik}-B_{ik}\sin {\theta }_{ik}\right)U_i{U}_{er}^{dc}\\ P_{loss,i}^{DAB,dc}=0.02P_{con}=0.02\left({\displaystyle \frac{n{U}_{DC}{U}_{dc}}{R_k}}-{\displaystyle \frac{U_{dc}^2}{R_k}}\right)\end{array}$$

(8)

The EER is a device in the new type of distribution system that facilitates power distribution and coordination. During the design process, it is essential to designate one port as a slack port to effectively manage the power losses within the EER.

The control methods for EERs include AC port control and DC port control, with specific control method classifications shown in

Table 2. The control methods for EERs.

Port Type	Control Strategies
AC Port	PQ control (active power control uses fixed $P_i^{ac}$ control; reactive power control uses fixed $Q_i^{ac}$ control)
	PV control (active power control uses fixed $P_i^{ac}$ control; reactive power control uses fixed $U_i$ control)
DC Port	Fixed DC voltage $U_{DC}$ control
	Fixed DC active power $P_{DC}$ control
	Droop control

.

In this paper, except for the power balance port, the AC ports use PQ control strategies, while the DC ports use either fixed DC active power control or droop control strategies [28]. The following sections provide an introduction to these three strategies.

(1) PQ control

When the AC port uses PQ control, it indicates that the active power $P_i^{ac}$ and reactive power $Q_i^{ac}$ at the connection between the AC/DC converter and the AC grid remain constant. During power flow calculation iterations, this can be equivalently represented as a PQ node, with the control equations as follows:

$$\{\begin{array}{l}{P}_{load}=P_i^{ac}\\ Q_{load}=Q_i^{ac}\end{array}$$

(9)

In the equation, $P_{load}$ represents the active power of the equivalent load at the AC port of the power router, and $Q_{load}$ represents the reactive power of the equivalent load at the AC port of the power router.

(2) Fixed DC active power $P_{DC}$ control

When the DC port uses fixed DC active power control, the active power $P_{DC}$ at the DC port is a known quantity, while the voltage $U_{DC}$ is the quantity to be determined.

(3) Droop control

When the DC port uses droop control, the relationship between the voltage $U_{DC}$ and the active power $P_{DC}$ at the DC port of the power router can be expressed as follows:

$$P_{DC}={\displaystyle \frac{1}{k_{dp}}}\left(U_{DC}-U_{DC}^{ref}\right)+P_{DC}^{ref}$$

(10)

In the equation, $U_{DC}^{ref}$ and $P_{DC}^{ref}$ are the parameter set values at the DC port, and the negative real number $k_{dp}$ represents the droop coefficient of the DC node.

3. Hybrid Power Flow Calculation Model Construction

Power flow calculation is a fundamental tool in electric power systems for analyzing and determining key parameters such as voltage, current, and power at various nodes within the network [29]. A schematic diagram of a hybrid distribution network containing multiple EERs is shown in Figure 5, where the exchange of power between AC and DC distribution networks occurs through $n$ EERs to achieve the mutual assistance of power supply in various regions of the distribution network [30].

The hybrid distribution network model, which incorporates multiple EERs, can be partitioned into three components: the AC distribution network, the DC distribution network, and the EERs, as illustrated in Figure 5. Subsequently, distinct power flow calculation methodologies are proposed for each component to establish robust power flow calculations in distribution networks encompassing EERs.

3.1. AC Distribution Network Power Flow Calculation

The power flow calculation of the AC distribution network involves solving a set of nonlinear algebraic equations [31]. The Newton–Raphson method is widely used for its rapid convergence and high accuracy in addressing nonlinear equation problems. Therefore, this paper employs the Newton–Raphson method for the iterative calculation of the AC distribution network. By employing stepwise linearization, the Newton–Raphson method transforms nonlinear equations into linear equation problems that are solved iteratively. Its iterative equations can be expressed as follows [32]:

$$\{\begin{array}{l}\mathsf{\Delta }F\left(X^{(k)}\right)=-J\mathsf{\Delta }{X}^{(k)}\hfill \\ X^{(k+1)}=X^{(k)}+\mathsf{\Delta }{X}^{(k)}\hfill \end{array}$$

(11)

In the formula, $X$ represents the state variables, $\mathsf{\Delta }X$ represents the correction to the state variables, $J$ represents the Jacobian matrix, and $\mathsf{\Delta }F\left(X\right)$ represents the deviation of the n-dimensional equations.

The equation representing the power of AC nodes in the hybrid distribution network [33], which includes EERs (excluding the balance node), can be represented as follows:

$$\{\begin{array}{l}{P}_i=U_i{\displaystyle \sum _{j=1}^n}{U}_j\left(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij}\right)\hfill \\ Q_i=U_i{\displaystyle \sum _{j=1}^n}{U}_j\left(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij}\right)\hfill \end{array}$$

(12)

In the formula, $P_i$ and $Q_i$ represent the total active and reactive power connected to the load, as well as the EER at node i. $U_i$ and $U_j$ are the voltage magnitudes at nodes i and j, respectively. Furthermore, ${\theta }_{ij}$ is the phase angle difference between nodes i and j, while $G_{ij}$ and $B_{ij}$ denote the conductance and susceptance between nodes i and j, respectively [34,35,36].

By combining Equations (11) and (12), it can be seen that the power deviation equation and the Jacobian matrix for the node can be represented as follows [37]:

$$\{\begin{array}{l}\mathsf{\Delta }{P}_i=P_i-U_i{\displaystyle \sum _{j=1}^n}{U}_j\left(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij}\right)\hfill \\ \mathsf{\Delta }{Q}_i=Q_i-U_i{\displaystyle \sum _{j=1}^n}{U}_j\left(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij}\right)\hfill \end{array}$$

(13)

$$\left[\begin{array}{c}\mathsf{\Delta }P\\ \mathsf{\Delta }Q\end{array}\right]=-\left[\begin{array}{cc}H& N\\ K& L\end{array}\right]\left[\begin{array}{c}\mathsf{\Delta }\theta \\ \mathsf{\Delta }U/U\end{array}\right]$$

(14)

In the formula, $\left[\begin{array}{cc}H& N\\ K& L\end{array}\right]$ is the Jacobian matrix, as shown below:

$$\{\begin{array}{l}\{\begin{array}{l}{H}_{ij}={\displaystyle \frac{\partial P_i}{\partial {\theta }_j}}=-U_i{U}_j\left(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij}\right)\hfill \\ H_{ii}={\displaystyle \frac{\partial P_i}{\partial {\theta }_i}}=U_i\sum _{j\in i,j\ne i}{U}_j\left(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij}\right)\hfill \end{array};\{\begin{array}{l}{N}_{ij}=U_j{\displaystyle \frac{\partial P_i}{\partial U_j}}=-U_i{U}_j\left(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij}\right)\hfill \\ N_{ii}=U_i{\displaystyle \frac{\partial P_i}{\partial U_i}}=-U_i\sum _{j\in i,j\ne i}{U}_j\left(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij}\right)-2U_i^2{G}_{ii}\hfill \end{array}\\ \{\begin{array}{l}{K}_{ij}={\displaystyle \frac{\partial Q_i}{\partial {\theta }_j}}=U_i{U}_j\left(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij}\right)\hfill \\ K_{ii}={\displaystyle \frac{\partial Q_i}{\partial {\theta }_i}}=-U_i\sum _{j\in i,j\ne i}{U}_j\left(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij}\right)\hfill \end{array};\{\begin{array}{l}{L}_{ij}=U_j{\displaystyle \frac{\partial Q_i}{\partial U_j}}=-U_i{U}_j\left(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij}\right)\hfill \\ L_{ii}=U_i{\displaystyle \frac{\partial Q_i}{\partial U_i}}=-U_i\sum _{j\in i,j\ne i}{U}_j\left(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij}\right)+2U_i^2{B}_{ii}\hfill \end{array}\end{array}$$

In the formula, $j\in i$ represents the set of nodes connected to node i.

3.2. DC Distribution Network Power Flow Calculation

Assuming there are n nodes in the DC distribution network, the injected power $P_{dc,i}$ at the i-th node can be expressed as follows:

$$\{\begin{array}{l}{P}_{dc,i}=P_{con,i}+P_{DG,i}-P_{load,i}\\ P_{dc,i}=U_{dc,i}{\displaystyle \sum _{j=1}^n}{G}_{dc,ij}{U}_{dc,j}\end{array}$$

(15)

In the formula, $P_{DG,i}$ represents the power injected into node i by the distributed power source, $P_{load,i}$ denotes the load at node i, and $G_{dc,ij}$j signifies the element in the i-th row and j-th column of the DC distribution network node admittance matrix [38].

Based on the control methods of the DC port as discussed in Section 2.3, the nodes of the DC distribution network can be classified into three categories: P-nodes, droop nodes, and slack nodes.

Supposing that the initial h nodes of the DC distribution network are P-nodes, the power flow equation is as follows [39]:

$$\mathsf{\Delta }{P}_{dc,i}=P_{dc,i}^s-U_{dc,i}{\displaystyle \sum _{j=1}^n}{G}_{dc,ij}{U}_{dc,j} (i=1,2,\dots ,h)$$

(16)

In Equation (16), $\mathsf{\Delta }{P}_{dc,i}$ is the power deviation at the i-th node, and $P_{dc,i}^s$ is the power setpoint at the i-th node.

Assuming the nodes from (h + 1) to (n − 1) in the DC distribution network are considered droop nodes, their power flow equation is as follows:

$$P_{dc,i}^{ref}+P_{DG,i}-P_{load,i}-{\displaystyle \frac{1}{k_{dp,i}}}{U}_{dc,i}^{ref}=U_{dc,i}{\displaystyle \sum _{j=1}^n}{G}_{dc,ij}{U}_{dc,j}-{\displaystyle \frac{1}{k_{dp,i}}}{U}_{dc,i}$$

(17)

To simplify the formula, let the calculation related to the droop node be denoted as ${\beta }_{dp,i}^s$; then, we have the following:

$$P_{dc,i}^{ref}+P_{DG,i}-P_{load,i}-{\displaystyle \frac{1}{k_{dp,i}}}{U}_{dc,i}^{ref}=P_{dc,i}^{ref}+P_{DG,i}-{\beta }_{dp,i}^s;{\beta }_{dp,i}^s=-U_{dc,i}{\displaystyle \sum _{j=1}^n}{G}_{dc,ij}{U}_{dc,j}+{\displaystyle \frac{1}{k_{dp,i}}}{U}_{dc,i}$$

(18)

From the above equation, it is evident that the power flow equation for the droop node is as follows:

$$\mathsf{\Delta }{\beta }_{dp,i}={\beta }_{dp,i}^s+U_{dc,j}{\displaystyle \sum _{j=1}^n}{G}_{dc,jj}{U}_{dc,j}-{\displaystyle \frac{1}{k_{dp,i}}}{U}_{dc,i} (i=h+1,\dots ,n-1)$$

(19)

The power flow equations of the DC distribution network are represented by Equations (18) and (19). When combined with the Newton–Raphson method, the resulting iterative equations are as follows [30]:

$$\{\begin{array}{c}\mathsf{\Delta }F\left(x^{(k)}\right)=-J_{dc}^{(k)}\mathsf{\Delta }{x}^{(k)}\\ x^{(k+1)}=x^{(k)}+\mathsf{\Delta }{x}^{(k)}\end{array}$$

(20)

In the formula, $\mathsf{\Delta }F=[\mathsf{\Delta }{P}_{dc,1},\mathsf{\Delta }{P}_{dc,2},\dots ,\mathsf{\Delta }{P}_{dc,h},\mathsf{\Delta }{\beta }_{dp,h+1},\dots ,\mathsf{\Delta }{\beta }_{dp,n-1}{]}^{\mathrm{T}}$, $x={\left[U_{dc,1},U_{dc,2},\dots ,U_{dc,n-1}\right]}^{\mathrm{T}}$, k represents the iteration count, and $J_{dc}$ denotes the Jacobian matrix on the side of the DC distribution network, with elements composed of the following [22]:

$$\{\begin{array}{l}{J}_{ij}={\displaystyle \frac{\partial \mathsf{\Delta }{P}_{dc,i}}{\partial U_{dc,j}}}=-G_{dc,ij}{U}_{dc,i} i=1,2,\dots ,h\\ J_{ij}={\displaystyle \frac{\partial \mathsf{\Delta }{\beta }_{dp,i}}{\partial U_{dc,j}}}=G_{dc,ij}{U}_{dc,i} i=h+1,\dots ,n-1\\ J_{ii}={\displaystyle \frac{\partial \mathsf{\Delta }{P}_{dc,i}}{\partial U_{dc,i}}}=-\left({\displaystyle \sum _{j=1}^n}{G}_{dc,ij}{U}_{dc,j}+G_{dc,ii}{U}_{dc,i}\right) i=1,2,\dots ,h\\ J_{ii}={\displaystyle \frac{\partial \mathsf{\Delta }{\beta }_{dp,i}}{\partial U_{dc,i}}}=\left({\displaystyle \sum _{j=1}^n}{G}_{dc,ij}{U}_{dc,j}+G_{dc,ii}{U}_{dc,i}\right)-{\displaystyle \frac{1}{k_{dp,i}}} i=h+1,\dots ,n-1\end{array}$$

(21)

3.3. EER Power Flow Calculation

Assuming there are r EERs in the hybrid distribution network, their collective set S is as follows:

$$S=\left\{{\mathrm{ER}}_1,\dots ,{\mathrm{ER}}_t,\dots ,{\mathrm{ER}}_r\right\}$$

(22)

Considering the information provided in Section 2.3, where PQ control is employed for the AC ports, let us assume that the t-th EER ${\mathrm{ER}}_t$ has $M_t$ primary ports and $N_t$ secondary ports. The power exchange at port l with the AC grid is denoted by $P_{il}^t$ and $Q_{il}^t$. Incorporating Equation (1), the nodal power equation for port l is expressed as follows:

$$\{\begin{array}{l}\mathsf{\Delta }{P}_{il}^t=P_{il}^t-\left(G_l^t\cos {\theta }_{ik}^t+B_l^t\sin {\theta }_{ik}^t\right)U_{il}^t{U}_{kl}^t\hfill \\ \mathsf{\Delta }{Q}_{il}^t=Q_{il}^t+\left(B_l^t\cos {\theta }_{ik}^t-G_l^t\sin {\theta }_{ik}^t\right)U_{il}^t{U}_{kl}^t\hfill \end{array}$$

(23)

In the formula, $U_{il}^t$ and $U_{kl}^t$ are the voltage values at the external and internal AC nodes connected to port l of ${\mathrm{ER}}_t$, respectively. $\mathsf{\Delta }{P}_{il}^t$ and $\mathsf{\Delta }{Q}_{il}^t$ are the biases of the active and reactive power of ${\mathrm{ER}}_t$, while $Y_l=G_l+j{B}_l$ stands for the admittance, and $B_{sl}$ represents the port capacitance.

In Equation (23), $U_{kl}^t$ corresponds to the voltages at the primary and secondary sides of the EER port and can be expressed as follows:

$$\{\begin{array}{l}{U}_{kl}^t=m_l^t{U}_{da}\\ U_{kl}^t=m_l^t{U}_{db}\end{array}$$

(24)

In the formula, $U_{da}$ and $U_{db}$ represent the voltages on the primary and secondary sides, respectively.

$$\begin{array}{l}\mathsf{\Delta }{P}_l^t=U_{da}{\displaystyle \sum _{l=1}^{{\mathrm{M}}_t}\left(-m_l^t{m}_l^t{G}_l^t{U}_{da}+m_l^t\left(G_l^t\cos {\theta }_{ik}-B_l^t\sin {\theta }_{ik}\right)U_{il}^t\right)}\\ +U_{db}{\displaystyle \sum _{l=1}^{{\mathrm{N}}_t}\left(-m_l^t{m}_l^t{G}_l^t{U}_{db}+m_l^t\left(G_l^t\cos {\theta }_{ik}-B_l^t\sin {\theta }_{ik}\right)U_{il}^t\right)}+U_{da}{\displaystyle \sum _{h=1}^{K_t}\left(n_h^t{n}_h^t{U}_{da}-U_{dc}^t\right)/R_h^t}+U_{db}{\displaystyle \sum _{h=1}^{N_t}\left(n_h^t{n}_h^t{U}_{db}-U_{dc}^t\right)/R_h^t}\end{array}$$

(25)

In the formula, $U_{dc}$ is the voltage on the secondary side of the DC/DC converter, $n_h^t$ is the conversion ratio of the converter, and $R_h^t$ is its internal resistance representing the losses.

Combining the Newton–Raphson method, the power equation of the EER is solved [40], resulting in the iterative formula as follows:

$$\{\begin{array}{l}\mathsf{\Delta }{F}^t=-J^t\mathsf{\Delta }{X}^t;\mathsf{\Delta }{F}^t=\left[\mathsf{\Delta }{P}_{i,l}^t,\dots ,\mathsf{\Delta }{P}_{i,M_l+N_l-1}^t,\mathsf{\Delta }{Q}_{i,1}^t,\dots ,\mathsf{\Delta }{Q}_{i,M_l+N_t}^t,\mathsf{\Delta }{P}_l^t\right]\\ \mathsf{\Delta }{X}^t=\left[m_l^t,\dots ,m_{M_l+N_l}^t,{\theta }_l^t,\dots ,{\theta }_{M_l+N_l}^t\right]\end{array}$$

(26)

In the formula, $\mathsf{\Delta }{F}^t$ represents the residual vector of the power equation set for the EER, $\mathsf{\Delta }{X}^t$ denotes the variable vector of the power equation set for the EER, and $J^t$ signifies the Jacobian matrix of the power equation set for the EER. For ease of notation, let $\theta ={\theta }_{ik}$; then, the Jacobian matrix is as follows:

$$J^t=\left(\begin{array}{c}{\displaystyle \frac{\partial \left(\mathsf{\Delta }{P}_i^t\right)}{\partial m^t}} {\displaystyle \frac{\partial \left(\mathsf{\Delta }{P}_i^t\right)}{\partial {\theta }^t}}\\ {\displaystyle \frac{\partial \left(\mathsf{\Delta }{Q}_i^t\right)}{\partial m^t}} {\displaystyle \frac{\partial \left(\mathsf{\Delta }{Q}_i^t\right)}{\partial {\theta }^t}}\\ {\displaystyle \frac{\partial \left(\mathsf{\Delta }{P}_l^t\right)}{\partial m^t}} {\displaystyle \frac{\partial \left(\mathsf{\Delta }{P}_l^t\right)}{\partial {\theta }^t}}\end{array}\right)$$

(27)

The process of differentiating the residual vector of the equation set for the EER yields the following expressions:

$$\{\begin{array}{l}{\displaystyle \frac{\partial \left(\mathsf{\Delta }{P}_i^t\right)}{\partial m^t}}=-\left(G^t\cos {\theta }^t+B^t\sin {\theta }^t\right)U_i^t{U}_{er}^{dc};{\displaystyle \frac{\partial \left(\mathsf{\Delta }{P}_i^t\right)}{\partial {\theta }^t}}=-m^t\left(-G^t\sin {\theta }^t+B^t\cos {\theta }^t\right)U_i^t{U}_{er}^{dc}\\ {\displaystyle \frac{\partial \left(\mathsf{\Delta }{Q}_i^t\right)}{\partial m^t}}=\left(B^t\cos {\theta }^t-G^t\sin {\theta }^t\right)U_i^t{U}_{er}^{dc};{\displaystyle \frac{\partial \left(\mathsf{\Delta }{Q}_i^t\right)}{\partial {\theta }^t}}=-m^t\left(B^t\sin {\theta }^t+G^t\cos {\theta }^t\right)U_i^t{U}_{er}^{dc}\end{array}$$

(28)

$$\{\begin{array}{l}{\displaystyle \frac{\partial \left(\mathsf{\Delta }{P}_l^t\right)}{\partial m^t}}=U_{da}{\displaystyle \sum _{l=1}^{{\mathrm{M}}_t}\left(-2m_l^t{G}_l^t{U}_{da}+\left(G_l^t\cos {\theta }^t-B_l^t\sin {\theta }^t\right)U_{il}^t\right)}+U_{db}{\displaystyle \sum _{l=1}^{{\mathrm{N}}_t}\left(-2m_l^t{G}_l^t{U}_{db}+\left(G_l^t\cos {\theta }^t-B_l^t\sin {\theta }^t\right)U_{il}^t\right)}\\ {\displaystyle \frac{\partial \left(\mathsf{\Delta }{P}_l^t\right)}{\partial {\theta }^t}}=U_{da}{\displaystyle \sum _{l=1}^{{\mathrm{M}}_t}{m}_l^t\left(-G_l^t\sin {\theta }^t-B_l^t\cos {\theta }^t\right)U_{il}^t}+U_{db}{\displaystyle \sum _{l=1}^{{\mathrm{N}}_t}{m}_l^t\left(-G_l^t\sin {\theta }^t-B_l^t\cos {\theta }^t\right)U_{il}^t}\end{array}$$

(29)

Subsequently, the initial conditions for the system can be specified, and Equations (24)–(29) can be combined to perform power flow analysis for the EER.

4. The Power Flow Calculation for AC/DC Hybrid Distribution Networks Considering EERs

The unified method and alternating iterative method are commonly utilized for calculating power flow in hybrid AC/DC distribution networks [41]. The inclusion of the EER model in the unified solution method significantly amplifies the complexity of the Jacobian matrix and prolongs computation time. Conversely, although employing the alternating iterative method may increase iteration count, it simplifies the Jacobian matrix, resulting in reduced computational time. Therefore, this paper proposes a method for the power flow calculation of AC/DC hybrid distribution networks considering EERs based on the alternating iterative method [42].

The first step is to input all the original data of the system, including the parameters related to the generator units, distributed power sources, branches, nodes, loads, and EERs of the AC/DC distribution network [43]. The second step is to divide and number the AC/DC distribution network according to voltage types and levels [44]. The third step is to define the reference directions for P and Q; in this paper, the direction of power injected into the EER from the power grid is set as the positive direction. The fourth step entails initializing the EER by configuring port control methods and setting initial values. PQ control is employed for the AC ports of the EER, while constant P control is adopted for DC ports. Then, $P_i$ and $Q_i$ for the power exchange at the AC ports are determined, and $P_i$ for the power exchange at the DC ports is determined accordingly. Step five involves performing power flow calculations on the DC distribution network. Lastly, the internal parameters of the AC ports within the EER are initialized.

The initial active power of the slack port can be expressed as follows:

$$P_{i,{\mathrm{M}}_t+N_t}^t=-{\displaystyle \sum _{l=1}^{M_t+N_t-1}}{P}_{il}^t-{\displaystyle \sum _{h=1}^{K_t+T_t}}{P}_{dc,h}^t$$

(30)

Based on Equations (1) and (3), the internal parameters of the EER are determined. To simplify the calculation, the conductance is neglected. The initial values can be expressed as follows:

$${\theta }_0^t=\arctan \left({\displaystyle \frac{-P_i^{ac,t}}{Q_i^{ac,t}+\left(B_{ik}^t+B_{si}^t\right)U_i^t{U}_i^t}}\right)$$

(31)

$$m_0^t={\displaystyle \frac{P_i^{ac,t}}{B_{ik}^t{U}_i^t{U}_{er}^{dc,t}\sin {\theta }_0^t}}$$

(32)

The seventh step requires the execution of a power flow computation for the AC distribution grid. If convergence is not achieved, the power of the AC distribution grid will be adjusted according to the active and reactive power levels at the EER ports. Supposing that node i of the AC distribution network is connected to port k of the EER, then the equivalent output active power $P_s$ and reactive power $Q_s$ at node i of the AC distribution network can be formulated as follows [45]:

$$\{\begin{array}{l}{P}_s=P_i+P_i^{ac,t}\hfill \\ Q_s=Q_i+Q_i^{ac,t}\hfill \end{array}$$

(33)

The eighth step involves conducting power flow calculation for the EER, solving for the variables $m_l^t$ and ${\theta }_l^t$ at the AC port of the EER, and updating the active power $P_{i,{\mathrm{M}}_t+N_t}^t$ at the slack port to $P_{i,{\mathrm{M}}_t+N_t}^{t\_new}$. Then, the variables are computed for all EERs, and the active power at their respective slack ports is updated.

Finally, the variation in active power injected into all slack ports of the EERs after the iteration is computed. If it fails to meet the required value, we proceed to step seven; if the requirement is fulfilled, the outcomes are presented, and the computation is terminated.

$$\max \left\{abs\left(P_{i,M_1+N_1}^{t\_\mathrm{new}}-P_{i,M_1+N_1}^1\right),\dots ,abs\left(P_{i,M_t+N_t}^{t\_\mathrm{new}}-P_{i,M_t+N_t}^t\right),\dots ,abs\left(P_{i,M_n+N_n}^{n\_\mathrm{new}}-P_{i,M_n+N_n}^n\right)\right\}<\epsilon$$

(34)

The process of power flow calculation for AC/DC hybrid distribution networks, considering EERs, is shown in Figure 6.

5. Case Study Analysis

To verify the accuracy and applicability of the AC/DC hybrid power flow calculation method considering EERs introduced in this study, the IEEE 69-node distribution network was adjusted to form a medium- and low-voltage AC/DC distribution system interconnected by EERs [46,47], as depicted in Figure 7. The constructed system comprises a 10 kV DC distribution network, a 10 kV AC distribution network, a 750 V DC distribution network, and a 380 V AC distribution network, which serves as the target for configuring EERs. By varying the EER configuration within the medium- and low-voltage AC/DC distribution system, case analyses are conducted under different scenarios to verify the applicability of the proposed AC/DC hybrid power flow calculation method that accounts for multiple ports of EERs and multiple EERs.

5.1. The Verification of Power Flow Calculation for Multi-Port EERs

Power flow calculation is predicated on instantaneous power values, and the treatment of distributed power sources aligns with that of loads [48,49]. Therefore, in the given case study, distributed power sources are categorized as loads, leading to the expansion of the original four-port EER to a six-port EER. The modified case is illustrated in Figure 8.

In this example case with a six-port EER, it is assumed that ports 2, 3, 4, 5, and 6 have an excess of power. All the surplus power is compensated to port 1 in order to alleviate the supply pressure on the 10 kV AC distribution network at this point. According to Formulas (21)–(23), the magnitude of the equivalent modulation coefficient $m$ is related to the impedance parameters, meaning that the port capacity can be reflected through the impedance parameters. Simulations have shown that when the transmission power of each port is equal to or less than 50 kW, the equivalent modulation coefficient $m$ can be maintained between 0 and 1. Therefore, in this example case, the port capacity is determined to be 50 kVA.

We assume that the nodes connected to ports 1 and 2 of the EER both lack 30 kW of active power, while the nodes connected to ports 5 and 6 can provide 10 kW and 50 kW of active power, respectively, in order to facilitate power sharing within the medium- and low-voltage AC/DC distribution network. The connection and power mutual aid situation of multiple multi-port EERs are shown in

Table 3. The power mutual aid situation of multi-port EERs.

Port Number	Connected Node	Control Method	$\mathit{R}\left(\mathit{p}.\mathit{u}.\right)$	$\mathit{X}\left(\mathit{p}.\mathit{u}.\right)$	${\mathit{X}}_{\mathit{s}\mathit{i}}\left(\mathit{p}.\mathit{u}.\right)$	${\mathit{P}}_{\mathit{i}}\left(\mathit{k}\mathit{W}\right)$	${\mathit{Q}}_{\mathit{i}}\left(\mathit{k}\mathit{V}\mathit{a}\mathit{r}\right)$
1	10 kV AC Node 5	PQ Control	0.0276	0.0251	7.1617	−30	0
2	10 kV AC Node 10	PQ Control	0.0276	0.0251	7.1617	−2	19
3	10 kV AC Node 15	PQ Control	0.0276	0.0251	7.1617	0	0
4	750 V DC Node 6	Constant P Control	0.0578	-	-	8	0
5	380 V AC Node 4	PQ Control	0.0276	0.0251	7.1617	30	10
6	10 kV DC Node 4	Constant P Control	0.0355	-	-	122	0

.

Using the power flow calculation method proposed in this paper, the calculation results for the six-port EER are shown in

Table 4. The calculation results of the six-port EER.

Port Number	${\mathit{P}}_{\mathit{i}}\left(\mathit{k}\mathit{W}\right)$	${\mathit{Q}}_{\mathit{i}}\left(\mathit{k}\mathit{V}\mathit{a}\mathit{r}\right)$	${\mathit{U}}_{\mathit{i}}\left(\mathit{p}.\mathit{u}.\right)$	${\mathit{U}}_{\mathit{e}\mathit{r}}^{\mathit{d}\mathit{c}}\left(\mathit{p}.\mathit{u}.\right)$	$\mathit{m}$	${\mathit{\theta }}_{\mathit{k}}(°)$
1	−30	0	0.9997	1.0000	0.5572	−46.7610
2	−30	0	0.9889	1.0003	0.5631	−46.7582
3	0	0	0.9786	1.0000	0	0
4	0	0	0.9999	-	-	-
5	10	0	0.9999	0.9995	0.8883	9.0150
6	50	0	0.9999	-	-	-

:

According to Table 4, it is observed that when the mutual aid power between ports remains below their rated capacity, the equivalent modulation coefficient $m$ falls within the range of 0 to 1, thereby meeting the requirements. On the other hand, in order to control power flow direction between ports and connected nodes, positive or negative values of control ${\theta }_k$ are employed for the EER. For example, when ports 1 and 2 supply power to the connected nodes, the corresponding ${\theta }_k$ for the ports is negative, whereas port 5 absorbs power from the connected node, resulting in the positive corresponding ${\theta }_k$.

To validate the precision of the power flow calculation suggested in this study, initial calculations were performed using matpower in MATLAB, following the methodology described herein to develop a power flow program. Subsequently, separate power flow analyses were conducted on the 10 kV DC distribution network, 10 kV AC distribution network, 750 V DC distribution network, and 380 V AC distribution network. Ultimately, a comparative analysis of the results demonstrated that the maximum error in the power flow calculation for the AC/DC distribution network was within 0.05%. Consequently, it can be concluded that the proposed AC/DC distribution network power flow calculation method presented in this article is both accurate and effective.

5.2. The Validation of Power Flow Calculation for Multi-Port and Multi-Quantity EERs

The aforementioned medium- and low-voltage AC/DC distribution system utilizes a four-port EER and a six-port EER to facilitate power mutual assistance within the distribution system.

The modified example case is shown in Figure 9:

The network parameters of the EERs are consistent with Table 3, thus indicating a power transmission capacity of 50 kVA for each port. In accordance with Figure 9, the connection method and power mutual aid situation between the multi-port multi-quantity EERs and the medium- and low-voltage AC/DC distribution network are shown in

Table 5. Connection and power mutual aid situation of multi-port and multi-quantity EERs.

EER Number	Port Number	Connected Node	Control Method	${\mathit{P}}_{\mathit{i}}\left(\mathit{k}\mathit{W}\right)$	${\mathit{Q}}_{\mathit{i}}\left(\mathit{k}\mathit{V}\mathit{a}\mathit{r}\right)$
EER1	1	380 V AC Node 3	PQ Control	10	0
	2	10 kV DC Node 6	Constant P Control	25	0
	3	750 V DC Node 2	Constant P Control	10	0
	4	10 kV AC Node 11	PQ Control	−45	0
EER2	1	10 kV AC Node 33	PQ Control	−40	0
	2	10 kV AC Node 31	PQ Control	−40	0
	3	10 kV DC Node 10	Constant P Control	35	0
	4	10 kV AC Node 25	PQ Control	35	0
	5	750 V DC Node 11	Constant P Control	0	0
	6	380 V AC Node 8	PQ Control	10	0

:

During the process of power mutual aid between ports and nodes, considering that the internal part of the EER consists of a DC bus with active power flow being predominant, the reactive power mutual aid is set to 0 in Table 5. The subsequent analysis employs the power flow calculation method proposed in this paper, and the calculation results for multi-port and multi-quantity EERs are shown in

Table 6. EER calculation results.

EER Number	Port Number	${\mathit{U}}_{\mathit{i}}\left(\mathit{p}.\mathit{u}.\right)$	${\mathit{U}}_{\mathit{e}\mathit{r}}^{\mathit{d}\mathit{c}}\left(\mathit{p}.\mathit{u}.\right)$	$\mathit{m}$	${\mathit{\theta }}_{\mathit{k}}(°)$
EER1	1	0.9999	1.0000	0.8879	9.0164
	2	0.9999	-	-	-
	3	0.9999	-	-	-
	4	0.9882	1.0265	0.8237	−47.0761
EER2	1	0.9882	1.000	0.7516	−46.9965
	2	0.9944	1.0003	0.7467	−47.0031
	3	0.9999	-	-	-
	4	0.9748	0.9996	0.6669	132.8134
	5	0.9999	-	-	-
	6	0.9994	0.9997	0.8876	9.0267

:

By combining Table 4 and Table 6, it becomes evident that the calculation parameters for the EER losses were determined, thus enabling the computation of EER power loss. Additionally, the port equivalent modulation coefficient $m$ and the corresponding angle ${\theta }_k$ remained consistent in both multiple EER and single EER power flow calculations. Thus, it is evident that the AC/DC hybrid distribution network power flow calculation considering the proposed EER is accurate and effective.

6. Conclusions

In response to the challenges faced by traditional distribution networks during the transformation and upgrading process of new-type power systems, this paper proposes a method for accurately calculating the power flow in AC/DC hybrid distribution networks considering EERs based on the alternating iterative approach. Finally, through the verification of power flow calculations involving multi-port and multi-quantity EERs, the following conclusions are drawn:

(1) The proposed AC/DC hybrid distribution network power flow calculation method considering EERs exhibits high accuracy. Compared to the results obtained from MATPOWER in MATLAB, the error margin is within 0.05%.

(2) The examples demonstrate the adaptability of the AC/DC power flow calculation method considering EERs, which can effectively accommodate multi-port and multi-quantity EER configurations.

(3) EERs can significantly advance the development of the energy internet, with promising prospects for optimal configuration in medium- and low-voltage AC/DC distribution networks.

In future work, further research will focus on optimizing EER configurations in relation to grid partitioning.