In the context of “carbon peak” and “carbon neutrality,” the penetration of distributed generation technology in the power grid is gradually increasing, and microgrids have become an effective form of utilizing distributed generation technology (Song et al., 2021; Zou et al., 2022). Hybrid AC/DC microgrids, combining the advantages of both AC and DC microgrids, have become an important direction in the development of microgrid technology. However, there exist complex coupling relationships between AC and DC within the system. Therefore, how to systematically analyze AC/DC microgrids and ensure their safe and stable operation has become a focal research area for scholars worldwide (Liu K. et al., 2019; Hameed et al., 2019; Yang et al., 2019).

Power flow analysis, as one of the fundamental tools for microgrid analysis, its mathematical essence involves solving a set of multivariate nonlinear equations through iterative computations to determine parameters such as voltage, phase angle, and power at various nodes (or buses) within the grid. Power flow analysis is employed to ascertain load distribution, voltage stability, line power losses, and flow distribution of components such as generators, transformers, and transmission lines within the power system (Guoqing et al., 2017; Pengfei, 2022), plays a crucial role in microgrid planning, stability calculations, and fault analysis (Bajpai, 2023; Heidary et al., 2023; Huang et al., 2023; Li et al., 2023). Currently, commonly used power flow calculation methods for AC/DC microgrids include the unified iteration method (UIM) and the alternate iteration method (Nejabatkhah et al., 2019; Song et al., 2023). The alternate iteration method separately solves power flow for AC and DC subsystems, providing fast calculation speeds but relatively lower convergence accuracy. On the other hand, the unified iteration method extends the dimension of the Jacobian matrix by introducing DC variables, allowing for the joint solution of variables in AC and DC systems. However, this increases the order of the Jacobian matrix, requiring matrix decomposition in each iteration, thereby adding to the overall computation time (Wang et al., 2018). Therefore, reducing computational complexity and time in the power flow calculation process while ensuring accuracy has become an important aspect of research on power flow calculation in hybrid AC/DC microgrids.

To enhance the efficiency of AC/DC microgrid power flow calculations, related research has primarily focused on improving existing methods. Reference (Liu et al., 2021) proposed a control mode for distributed grids participating in voltage and frequency regulation based on the Newton-Raphson method. On this basis, it introduced a sequential algorithm framework for handling coupling relationships between AC and DC microgrids. Using an improved Newton-Raphson sub-algorithm, it iteratively solved power flow for hybrid AC/DC microgrids. Reference (Ju et al., 2022) presented a fully Distributed Power Flow (DPF) method, transforming nonsmooth constraints into smooth functions. It employed a two-level Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method with second-order convergence speed to convert the DPF problem into a distributed step-size optimization problem. By exchanging microgrid boundary information, it achieved accurate power flow results and improved convergence speed and accuracy through the second-level step-size optimization. Reference (Chen et al., 2017) established a steady-state power flow model for a Droop-type distributed power source and AC/DC inverter. It applied a sequence component conversion based on voltage symmetry at the grid connection point and used a sequence current compensation method to decouple the AC subgrid into three-sequence networks, solving them in parallel and further reducing the solution capacity.

While the above literature has improved computational accuracy and speed through enhancements to existing methods, it does not address the impact of DC system characteristics on power flow calculation equations. To address this gap, this paper first comprehensively considers DC system characteristics and proposes an AC/DC microgrid power flow algorithm based on the unified iteration method. This resolves the issues of long computation times and high computational capacity associated with the unified iteration power flow calculation method. Subsequently, through simulation experiments on the IEEE 11-node system, the effectiveness and high computational efficiency of the proposed algorithm are validated. The study identifies the factors and extent of error generation under different conditions. Finally, experimental verification on a 30 kW load and 70 kW load is conducted on the experimental platform, confirming the feasibility of the improved algorithm.

In an AC/DC microgrid, which includes both AC and DC lines, the selection of DC lines leads to variations in system structure and operational modes. Consequently, there are differences in the equivalent models and DC equations.

Taking a two-terminal bipolar DC transmission system as an example, the selected DC line model is systematically modeled (Yang et al., 2021; Zhu et al., 2022), resulting in the equivalent circuit of the DC system as illustrated in Figure 1:

From the equivalent circuit shown in Figure 1, the expressions for the main parameters of the DC line can be obtained as follows: I d = 3 2 E r ⁡ cos ⁡ α / π − 3 2 E i ⁡ cos ⁡ β / π d x r + R 1 + d x i (1) P d r = U d r I d P d i = U d i I d = P d r − I d 2 R 1 (2) U d r = 3 2 E r ⁡ cos ⁡ α / π − I d d x r U d i = 3 2 E i ⁡ cos ⁡ β / π + I d d x i U d r = U d i + I d R i (3)

Eqs 1–3 form the main parameter equations for the two-terminal bipolar DC transmission system.

For a hybrid AC/DC system model with a total of n nodes, let the number of DC nodes be n d , and the number of AC nodes be n a = n − n d , To facilitate the construction of correction equations, the nodes of the system are numbered as follows: the first n a nodes are labeled as AC nodes, and the remaining n d nodes are labeled as DC nodes.

For a pure AC system with n a nodes, the power equations and correction equations in the polar coordinate form of the Newton-Raphson method are expressed as Eqs 4, 5, respectively. Δ P i = P i − U i ∑ j = 1 n U j G i j ⁡ cos ⁡ δ i j + B i j ⁡ sin ⁡ δ i j   i = 1 , 2 , . . . , n a Δ Q i = Q i − U i ∑ j = 1 n U j G i j ⁡ sin ⁡ δ i j − B i j ⁡ cos ⁡ δ i j   i = 1 , 2 , . . . , n a (4) Δ P a Δ Q a = − H a a N a a M a a L a a Δ δ a Δ U a / U a = − J Δ δ a Δ U a / U a (5) Where the subscript ‘ a ’ represents AC variables; Δ P a = Δ P 1   Δ P 2 ⋯ Δ P n a T , Δ Q a = Δ Q 1   Δ Q 2 ⋯ Δ Q n a T , H a a , N a a , M a a , L a a are the sub-matrix of partial derivatives of the quantity to be corrected with respect to the correction variables, such as H a a = ∂ Δ P a / ∂ δ a , N a a = ∂ Δ P a / ∂ U a × U a 。

The power equations of the UIM are established based on the power equations of the AC system, with the inclusion of DC variables (Peng et al., 2017; Liu K et al., 2019; Lin et al., 2020; Wang et al., 2023). Combining the proposed DC model, the power equation for the hybrid AC/DC system is obtained by introducing DC variables into Eq. 4, as shown in Eq. 6: Δ P i = P i − U i ∑ j = 1 n U j G i j ⁡ cos ⁡ δ i j + B i j ⁡ sin ⁡ δ i j ± U d k I d k , i = n d + k , k = 1 , 2 , ⋯ n d Δ Q i = Q i − U i ∑ j = 1 n U j G i j ⁡ sin ⁡ δ i j − B i j ⁡ cos ⁡ δ i j ± U d k I d k ⁡ tan ⁡ φ k , i = n d + k , k = 1 , 2 , ⋯ n d (6) Where the positive and negative signs represent the inverter and rectifier, respectively.

Considering the presence of DC variables in the mixed AC/DC system, in Eq. 5, active power and reactive power imbalance terms for the DC nodes, as well as phase difference and voltage difference, are introduced. From Eq. 6, it can be observed that the correction equations introduce new variables U d k , I d k , φ k . At this point, the number of variables exceeds the number of equations. According to the theory of the boundedness of solutions in a linear space, it is necessary to supplement the correction equations with new equations to ensure that the number of equations is greater than or equal to the number of variables.

Combining Eqs 1–3, supplement the parameter equation Δ Z p and control equation Δ Z p for the DC system in the quantity to be corrected, and express them uniformly as Δ Z .

Rearranging Eq. 3 and forming the parameter equation Δ Z p for the DC system is shown in Eq. 7 (Chen et al., 2019; Lee et al., 2020): Δ Z p = U d r − 3 2 E r ⁡ cos ⁡ α / π + I d d x r = 0 U d i − 3 2 E i ⁡ cos ⁡ β / π − I d d x i = 0 U d r − R i I d − U d i = 0 (7)

The main control equation for the DC system is expressed as Eq. 8: Δ Z c = P − P s = 0 β − β s = 0 α − α s = 0 I d − I s = 0 U d − U s = 0 K r − K s = 0 K i − K s = 0 (8) Where the variables with subscript‘s’ represent the system rated values; Δ Z c is determined based on the specific control method.

Due to the increase in the quantity to be corrected on the left side of the correction equation, and with only phase difference and voltage difference in the correction discrepancy, it is still not possible to solve the correction equation. Considering the inclusion of Eqs 7, 8 in the quantity to be corrected, and incorporating the characteristic parameters of the DC system, the correction discrepancy for the DC system is expanded by introducing the correction variable Δ X . The specific expression is shown in Eq. 9. Δ X = Δ U d T   Δ I d T   Δ K d T   Δ W T   Δ Φ T T (9)

The specific expressions for each variable in Δ X are shown in Eq. 10: Δ U d = Δ U d 1   Δ U d 2   Δ U d 3 ⋯ Δ U d n d T Δ I d = Δ I d 1   Δ I d 2   Δ I d 3 ⋯ Δ I d n d T Δ K T = Δ K T 1   Δ K T 2   Δ K T 3   ⋯ Δ K T n d T Δ W = Δ ⁡ cos ⁡ θ d 1   Δ ⁡ cos ⁡ θ d 2 ⋯ Δ ⁡ cos ⁡ θ d n d T Δ Φ = Δ φ 1   Δ φ 2   Δ φ 3 ⋯ Δ φ n d T (10)

Thus, the correction equation for the UIM in the mixed AC/DC system is: Δ P a Δ P d Δ Q a Δ Q d Δ Z = − H a a H a d N a a N a d 0 H d a H d d N d a N d d A M a a M a d L a a L a d 0 M d a M d d L d a L d d B 0 0 0 C D Δ δ a Δ δ d Δ U a / U a Δ U d / U d Δ X = − J ′ Δ δ a Δ δ d Δ U a / U a Δ U d / U d Δ X (11)

Among them, A = ∂ Δ P d / ∂ X ; B = ∂ Δ Q d / ∂ X ; C = ∂ Δ Z / ∂ U d × U d ; D = ∂ Δ Z / ∂ X ; Where the elements of the sub-matrices H , N , M , and L in J ′ are the same as those in Eq. 5.

Compared to the Jacobian matrix J in Eq. 5, the order of the augmented Jacobian matrix J ′ in Eq. 11 has increased by 5× n d . The increase in the order of the Jacobian matrix can complicate the process of solving the inverse matrix, potentially leading to singular values and causing non-convergence of the calculation results. Additionally, it occupies a considerable amount of computational memory and increases the power flow computation time, necessitating improvements.

In DC lines, the power factor is determined by factors cos ⁡ φ = 1 and Q = U I s i n φ . Meanwhile, only resistive loads are effective in DC circuits, and during power flow analysis, only steady-state changes need to be considered. Therefore, in DC, only active power consumption is considered, and the flow of reactive power is not taken into account. As a result, there is no need for iterative calculations for the variable Δ Q d in the quantity to be corrected.

At the same time, when transmitting electric energy in DC form, due to the special characteristics of DC power supply, there is no induction of reactance in the line during the energy transmission process, and there is no issue of voltage phase angle. For the DC portion in the mixed AC/DC system, there is no need to consider the phase deviation Δ δ d in the correction discrepancy, which means removing the variables Δ Q d and Δ δ d from Eq. 11. The resulting correction equation is shown in Eq. 12: Δ P a Δ P d Δ Q a Δ Z = − H a a N a a N a d 0 H d a N d a N d d A M a a L a a L a d 0 0 0 C D Δ δ a Δ U a / U a Δ U d / U d Δ X (12)

Eq. 12 represents the expression of the corrected equation after improvement, which is the correction equation of the improved unified iteration method (IUIM).

The power flow annotation for the modified IEEE 11-node system structure is shown in Figure 6.

From the power flow illustrated in Figure 6, it is evident that for the hybrid AC/DC system, nodes 3 and 5 involve issues regarding the synthesis and distribution of power between AC and DC systems. In this paper, certain DC parameters in the correction equations during DC power flow computation were removed, leading to relative errors between the actual values and the influx and distribution of DC power at the interface nodes between the DC and AC systems. Figure 7 shows the phase error curve of node voltages. As the electrical distance between nodes and the reference node increases, the phase error of node voltages gradually increases. In multi-node radial networks, an increase in the electrical distance between nodes and the reference node leads to a gradual increase in calculation errors. However, for microgrid systems with lower voltage levels, where the electrical distance between nodes is shorter, the errors generated by the improved algorithm are relatively small.

Comparing Eqs 11, 12, the deviation of active power and reactive power for a single iteration process at a certain node in the system is given by Eq. 13: Δ Δ P = ∂ Δ P a ∂ δ d + ∂ Δ P d ∂ δ d Δ δ d Δ Δ Q = Δ Q d + ∂ Q a ∂ δ d Δ δ d (13)

Analyze the impact of U i or U j and Δ δ d on Δ Δ P and conduct a quantitative analysis. It is observed that the expressions for the partial derivatives of Δ P a with respect to Δ δ d are consistent with those for Δ P d . For ease of analysis, ∂ Δ P a / ∂ δ d and ∂ Δ P d / ∂ δ d are unified as ∂ Δ P / ∂ δ d . Taking node voltage U i = U j , G i j + B i j = 5 + j 10 , we plot the error influence curves of Δ δ d and U i on ∂ Δ P / ∂ δ d , as shown in Figure 8, with all variables represented in per unit values. Combining the simulation results, the curves for active power error and node voltage error are plotted separately in Figures 9, 10.

From Figure 8, it can be observed that when U i remains constant, at lower values of Δ δ d , the numerical impact of Δ δ d on Δ Δ P is relatively small. As the value of Δ δ d increases, the impact on Δ Δ P gradually increases, and the degree of impact continues to grow without an upper limit. When Δ δ d remains constant, at lower values of U i , the numerical impact on Δ Δ P is relatively small. As the value of U i increases, the impact on Δ Δ P gradually increases, reaching a maximum degree of impact. The impact curve approximately exhibits a periodic function.

At lower voltage levels, Δ Δ P is primarily influenced by Δ δ d . As both Δ δ d and U i increase, Δ Δ P is influenced by both variables with similar degrees of impact. As Δ δ d continues to increase, Δ Δ P is primarily influenced by U i . In the example, where the voltage phase Δ δ d is relatively small, the impact of U i on Δ Δ P is more pronounced compared to Δ δ d . This is evident in nodes 3, 5, and 6 where the node voltages have relatively large errors, corresponding to peaks in the active power error curve.

Analyze the impact of Δ Q d and Δ δ d on Δ Δ Q , we plot the error influence curves of Δ Q d and Δ δ d on Δ Δ Q in Figure 11, with all variables represented in per unit values. Combining the simulation results, the reactive power error curve is plotted in Figure 12.

When Δ Q d remains constant, as the value of Δ δ d increases, the numerical impact on Δ Δ Q gradually increases, and the degree of impact continues to grow without an upper limit. When Δ δ d remains constant, Δ Q d consistently has a significant impact on Δ Δ Q . Combining the power flow diagram in Figure 8, for nodes 3 and 5, there is a relationship Δ Q = Δ Q a + Δ Q d . For lower values of Δ δ d , Δ Δ Q is primarily influenced by Δ Q d . As Δ δ d increases, Δ Δ Q is influenced by both variables with similar degrees of impact. As Δ δ d continues to increase, Δ Δ Q is primarily influenced by Δ δ d , and the degree of impact becomes more significant.

Observing Figure 12, it can be noted that there are significant errors in the reactive power on branches with the branch numbers 3, 4, 7, and 9. In the example, where the voltage phase Δ δ d is relatively small, Δ Δ Q is primarily influenced by Δ Q d . The lines with significant errors are mainly located at the connection points of DC and AC lines, specifically at nodes 3 and 5. In the iteration process, elements of Δ Q d in the correction equation for these two nodes were removed, leading to errors in the overall numerical values. Moreover, the closer the lines are to nodes 3 and 5, the larger the errors generated.

For systems with numerous parallel AC and DC lines in the network, the convergence and distribution of AC and DC power flows occur at the connection points. Ignoring Δ Q d can lead to significant errors in the iterative calculation of reactive power for the system, and the errors increase with more parallel branches. In microgrid systems with a common bus configuration, where there are fewer parallel branches, the overall error in the iterative calculation of reactive power is relatively small.

In summary, Δ Δ P is primarily influenced by the voltage phase Δ δ d and voltage magnitude U i . Δ Δ Q is mainly influenced by Δ δ d and the reactive power imbalance Δ Q d . When Δ δ d is small, Δ Δ P is primarily influenced by U i . When Δ δ d and U i are large, Δ Δ P is mainly influenced by Δ δ d , and the degree of influence is significant. Δ Δ Q is primarily influenced by Δ Q d . At the connection points of AC and DC lines within the mixed AC/DC system, relatively large errors in Δ Δ Q and node voltage phase can occur.

This paper, based on the characteristics of DC systems, simplifies the correction equations of the unified iteration method and proposes a power flow calculation model for hybrid AC/DC microgrids based on the improved unified iteration method. The following conclusions can be drawn through theoretical analysis and simulation/experimental verification:

1. By neglecting the flow of reactive power within the DC system and the influence of phase angles on the system, the Jacobian matrix in the correction equations of the unified iteration method is simplified and reduced in order to derive a power flow calculation method for hybrid AC/DC microgrids based on the unified iteration method. The case results indicate that the proposed improved algorithm improves computational speed by 80% compared to the Unified Iterative Method.