As introduced in many research studies and technical manuals, the loss of power in a power electronics converter can be described as a function of the output power. It has been studied in some research works. In the study by Yuan et al. (2021a), the converter loss is further approximated as a quadratic function by the output powers of DEUs. Then, the output active powers and reactive powers of the individual converter are dispatched by the concave allocation algorithm. In the study by Wang et al. (2010) and Teng et al. (2016), one converter-discarding strategy is used to adjust the number of operating converters. In this way, the overall efficiency of parallel-connected inverters is improved according to the demanding power. However, in these works, the potential for reducing converter loss has not been fully developed (Yuan et al., 2021b). Therefore, considering this in the control scheme is very important to minimize the power loss in the distribution network. For the AC microgrid with parallel DEUs, the output current of DEU flows through the corresponding converter and wire resistance to supply power to the load. Compared with a mesh network, there is no complex coupling relationship between output currents. As mentioned earlier, the distributed loss of a single DEU is a quadratic function. The sum of them is the total loss, which is naturally a concave function. Given the balance of energy supply and consumption, this loss model is one concave function. Through the on-line adjustment of current coefficients, the loss of the distribution network can be accurately minimized.

Inspired by the work on the concave optimization–based economical dispatch for generation cost minimization (Akbari-Dibavar et al., 2021), the concave optimization technique such as the dual ascent optimization method (DAOM) is applied in this work to solve the loss minimization problem. In recent years, the concave optimization algorithms, that is, Lagrange multiplier algorithm (Jiang et al., 2021c), dual decomposition algorithm (Jiang et al., 2021d), DAOM (Chen et al., 2017), and alternating direction method of multipliers (ADMM) (Chen and Yang, 2018) have attracted lots of attention for its wide implementations to solve problems in real systems, that is, parallel calculation, power dispatch, machine learning, and game theory. However, some practical problems for the implementation of power electronics should be considered. For example, the computation complexity of the applied algorithm should not affect the closed-loop PWM control of the MCU. By using a centralized data-exchange network, the Lagrange multiplier needs division operation and global data to obtain results (Jiang et al., 2021d), which reduces the scalability in large-scale AC microgrids. In the dual ascent algorithm, the division operation is necessary for current coefficient calculation. By adding a quadratic term of the constraint function, the convergence rate of ADMM is accelerated. However, such operation leads to an increase in computational complexity, which reduces its applicability in real-time high-frequency PWM control. For the DAOM, both Lagrange multiplier and current coefficients can be obtained through simple addition and subtraction iteration for implementation. Thereby, it is reasonable to select the DAOM for concave optimization.

After obtaining the optimal current coefficient, the three-phase converter shall adopt an appropriate active and reactive power distribution strategy. So far, different strategies have been proposed. To achieve accurate power distribution, the droop scheme regulates the output voltage and frequency to compensate for the imbalance among output powers from DEUs (Sun et al., 2017), (Loh et al., 2013). In the study by Sun et al. (2017), the adaptive voltage term is introduced by using active sharing error to regulate the bus voltages. Moreover, this droop scheme is combined with an integrator for power quality improvement (Loh et al., 2013). When inductive reactance plays a leading role in the line, the virtual inductance is generated by the active power dispatch term for frequency regulation (Morstyn et al., 2018).

In the hybrid microgrid, the power balancing control is achieved by one proper-designed interconnection control scheme (Wu et al., 2018). A multi-agent–based power sharing control is used to allocate the power flow to enhance voltage quality (Yang et al., 2020). The voltage error and frequency control are used to regulate the power distribution and bus voltage simultaneously (Yao et al., 2011). Vijay et al. (2021) used the consensus algorithm to adaptively regulate the bus voltage of DEUs according to the measured values of DEU nodes so as to achieve realize bus voltage restoration. So far, relatively little work has been carried out in considering the power loss as the control parameter. Generally, for the scenarios of long-distance AC networks, the wire inductance is large. Thereby, the active power-frequency control is usually applied by considering the inductive impedance. According to the different types of wire, the modified droop strategy has been designed and implemented in the case of pure inductance (Jiang et al., 2021e; Keyvani-Boroujeni et al., 2021; Huang et al., 2022) and pure resistance impedance (Luo et al., 2022), but few research studies explore the complex impedance of the system. In applications, the distance between different converter units is a little far, and the inductance and resistance wire impedance cannot be ignored through the parallel connection of different wires. In addition, due to the possibility of multi-feedback design, the impedance of the power converter is actually very complex. Due to the coupling between the active and reactive power, traditional control cannot achieve efficient power sharing.

As mentioned earlier, the distributed losses of one single DEU in the AC microgrid can be written as a quadratic function. As a combination, the total loss function of the AC microgrid can be obtained. As a multi-variable function, it is proven that the total loss is a concave function with respect to the current distribution coefficients. Furthermore, considering the supply-consumption balance, the proposed total loss model is an equality constrained concave function. In this work, the double ascent optimization method (DAOM) is designed to find the optimal current coefficient. By conducting the Lagrangian model, the optimal current coefficient can be iteratively calculated by the DAOM by simple addition and subtraction. In order to apply the current coefficient to the general PQ control structure, the calculation method of conversion between the current coefficient and power coefficient is designed. With the help of average reactive power, the active power distribution coefficients are obtained by using the voltages and currents of DEUs. For short-distance AC microgrids with a high resistance-to-reactance ratio (for example, R/X = 7.7), the conventional P-ω droop control and Q-E droop control are not suitable for power distribution among DEUs. In this work, an improved adaptive droop control strategy, composed of P-E droop control and Q-ω droop control, is proposed to dispatch the output active powers and reactive powers of DEUs. In this way, a multilayer control structure is proposed, including the DAOM in the top layer, P-E droop control and Q-ω droop control in the second layer, and voltage-current control in the lower layer. Simulation results verify the effectiveness of the proposed control strategy in reducing distribution losses.

As presented in Figure 2, the sketch map of the concave function is presented, where the concave function P _loss (N _i ) can be seen graphically. Here, the one concave optimization algorithm DAOM is selected for iterative optimization.

Based on (15), an augmented Lagrange function is defined as L ρ i ( N i , λ ) = ( a i + R i ) ( N i I tol ) 2 + b i N i | I tol | + c i + λ ( ∑ i = 1 n N i − 1 ) + ρ 2 ( ∑ i = 1 n N i − 1 ) 2 , (16) where λ is the Lagrange multiplier and λ ≠ 0, ρ is a coefficient. As given in the study by Jiang et al. (2021d) and Boyd and Vandenberghe (2004), all calculations for λ and optimal current allocation coefficients are completed in the central controller. With a large number of DEUs, the heavy calculation burden affects the scalability of the given Lagrange multiplier method. Instead of solving the function directly, the DAOM is proposed as an iterative optimization. The idea is to start at an initial guess, take a small step in the direction of the gradient, and repeat. Specifically, we use gradient ascent on the dual variables. The current coefficients can be updated as follows: N i k + 1 = arg min L ( N i k , λ k ) . (17)

The following equation should be satisfied as ∇ N i , λ L ( N i k , λ k ) = ( ∂ L ∂ N i k , ∂ L ∂ λ k ) = 0 (18)

which yields N i k + 1 = − b i | I tol | + λ k + ρ ( ∑ i = 1 n N i k − 1 ) 2 ( a i + R i ) ( I t ) 2 . (19)

Updating λ based on the residual of the linear constraint, the update rule in the central controller for λ can be expressed as follows: λ k + 1 = λ k + α ( ∑ i = 1 n N i k + 1 − 1 ) , (20) where α is a coefficient. The main advantage of the DAOM is the parallel execution of N _i updates. It is noteworthy that this inequality constraint will not change the convexity of the proposed loss model. Due to inequality constraints, some current coefficients will be set to constant values. Interestingly, the new loss function formed by the remaining DEUs is still a concave function with respect to the remaining N _i . Thereby, the DAOM can still be used for optimization.

As designed earlier, the current coefficient is obtained for loss minimization. Given the generally used PQ control structure, the current coefficient cannot be applied directly in the AC microgrid. Therefore, the current coefficient should be converged to the power distribution variables for power dispatch. Hence, the relationship between the output current and output power in a DEU is discussed for further control development. In an AC microgrid, the injection power is determined by the ratio of resistance and inductive reactance in the wire. As summarized in Table 1, in a short-distance AC microgrid, the injection active power is determined by the voltage amplitude of the DEU, and the injection reactive power is determined by the frequency of the DEU. The equivalent circuit of DEU _i is plotted in Figure 3: Z _i = R _i + jX _i is the wire impedance, θ _i is the wire impedance angle, E _i ∠φ _i is the terminal electromotive force of the converter, and V _bus∠0^◦ is the voltage of the common AC bus.

As given in Table 1, the injection power by DEU _i is given as P i ≈ V bus E i R i − V bus 2 R i ,   Q i ≈ − V bus E i R i ϕ i , (21)

As given in (21), the output active power of the converter is positively correlated with the amplitude of the terminal electromotive force. The output reactive power of the converter is negatively correlated with the operating frequency of the converter. In this way, one droop control is designed as E ref i = E nom − m i P i ,   m i = ( E max − E min ) / P max i , (22.1) ω ref i = ω nom + n i Q i ,   n i = ( ω max − ω min ) / Q max i , (22.2) where ω _refi and ω _nom are the reference value and nominal value of angle frequency, respectively, and E _refi and E _nom are the reference value and nominal value of angle frequency voltage amplitude, respectively. Meanwhile, the limits of angle frequency and voltage amplitude are also considered in (22): m _i and n _i are the droop coefficients, Q _maxi is the reactive power limit of the DEU _i , ω _min and E _min are the minimum values of angle frequency and voltage amplitude, and ω _max and E _max are the maximum values of angle frequency and voltage amplitude, respectively. According to the apparent power, the output current of the DEU _i is I i = 2 P i 2 + Q i 2 / E i = P i 2 + Q i 2 / V i , (23) where V _i is RMS values of voltage E _i ∠φ _i . Based on (23), it yields ( V i ) 2 ( N i I t o l ) 2 = ( N P i P t o l ) 2 + ( N Q i Q t o l ) 2 , (24.1) where P _tol and Q _tol are the total demanding powers. N _Pi and N _Qi are defined as the active power and reactive power coefficients of DEU _i , respectively. Thus, the average reactive power distribution is adopted as N Q i = 1 / n . (24.2)

Given the power generation limitation, the active power can be dispatched as N P i = { P min i / P t o l P i < P min i P max i / P t o l P i > P max i ( V i ) 2 ( N i I t o l ) 2 − ( Q t o l / n ) 2 / P t o l others (24.3)

Moreover, a secondary adaptive control layer is implemented as e P i = − ∑ i ≠ j n ( x P i − x P j ) , e Q i = − ∑ i ≠ j n ( x Q i − x Q j ) , (25) where x _Pi = P _i /N _Pi and x _Qi = Q _i /N _Qi . x _Pi and x _Qi are defined as the inter-media variables for power distribution. Then, the adaptive voltage and frequency terms are generated as E adp i = ( k E P i + k E I i / s ) e P i , ω adp i = ( k ω P i + k ω I i / s ) e Q i , (26) where k _Epi , k _EIi , k _ωPi , and k _ωIi are the gains of controllers. Thus, the secondary control is formed as E ref i = E nom − m i P i − E adp i , ω ref i = ω nom + n i Q i + ω adp i . (27)

As plotted in Figure 4, the proposed P-E and Q-ω control are presented in (a), and the diagram is given in (b). Based on the concave loss model, the optimal current distribution coefficients of each inverter are computed according to the DAOM. By using average reactive power sharing, the active power distribution coefficients are calculated. In addition, the output power of each DEU is calculated locally. Then, these coefficients are incorporated into the local droop controller to perform optimum power sharing. The reference values of angle frequency and voltage amplitude by adaptive droop control are further used for local voltage and current closed-loop control. In this way, the concave optimization–based current and power sharing for distribution loss minimization is achieved.

The convexity of the loss model is preliminarily verified by the exhaustive search method in the simulation model. By fixing the change step of the current coefficient, the new coefficient is continuously brought into the AC microgrid to obtain the loss value. For example, the DEU2 and DEU3 are selected to form an AC microgrid. Then, the distribution losses with respect to current coefficients under different load currents (8, 12, 16, and 20 A) are plotted in Figure 5A. The loss curves of another AC microgrid with DEU1 and DEU4 are also presented in Figure 5B. Due to the equality constraint, only one coefficient can be adjusted for an AC microgrid in the case of two DEUs. As can be seen, both eight curves in Figure 5 are conic with one minimum point.

Furthermore, two current coefficients can be regulated in the searching process for the three DEU-based AC microgrid under 30 A load current. Therefore, the system loss can be drawn as a three-dimensional surface. As given in Figure 6, four loss surfaces correspond to four different DEU combinations: DEU₁, DEU₂, and DEU₃ in Figure 6A, DEU₁, DEU₂, and DEU₄ in Figure 6B, DEU₁, DEU₃, and DEU₄ in Figure 6C, and DEU₂, DEU₃, and DEU₄ in Figure 6D). The contours of the losses are also projected onto the bottom surface. Apparently, both surfaces with openings upward are smooth, which are the typical concave surfaces.

For the system with four DEUs, the loss surface cannot be plotted directly with three adjustable variables. Therefore, we can fix the current coefficient value of one DEU so as the loss model degenerates into a three-dimensional surface. The total load current is 48 A. By fixing the one current coefficient at 0.1 in turn, four three-dimensional loss surfaces are obtained in Figure 7. Obviously, both surfaces with an opening upward are smooth, which are typical concave functions. For an AC microgrid with more DEUs, the convexity of the proposed loss model cannot be graphically illustrated. However, as shown in the previous figures, the theoretical proof has been provided in Section 2

Optimal control studies by the proposed DAOM and the P-E and Q-ω control are also presented based on the aforementioned analyzed AC distribution network. In case 1, the traditional control and the proposed control are adopted for the four DEUs in the periods from 0 to 3 s and 3–32 s, respectively. Figure 8 shows the waveforms of the voltages, output currents, current distribution coefficients, active powers, active power allocation coefficients, reactive powers, reactive power allocation coefficients for the DEUs in the AC microgrid, and the loss. At the initial condition, the current coefficients of DEUs are N ₁ = 0.33, N ₂ = 0.18, N ₃ = 0.21, and N ₄ = 0.28, and the output active power allocation coefficients of DEUs are N _P1 = 0.34, N _P2 = 0.17, N _P3 = 0.21, and N _P4 = 0.28. The output powers are P ₁ = 3.3 kW, P ₂ = 1.6 kW, P ₃ = 2.0 kW, and P ₄ = 2.75 kW and Q ₁ = Q ₂ = Q ₃ = Q ₄ = 0.94 kVar. These output power and current allocation rates are determined by the different wire impedances and voltage for DEUs. Meanwhile, b uses the converter loss coefficients and wire resistances in Tables 2 and 3, and the loss is calculated as 1097 W (converter loss is 812 W, and the wire loss is 285 W).

When the proposed DAOM is applied at 3 s, the current distribution coefficients, active power allocation coefficients, and reactive power allocation coefficients are continuously updated in a cycle of 1 s. At the same time, the distributed secondary control regulates the output active powers and reactive powers according to the updating power ratio. After iteration, the current distribution coefficients are optimized at N ₁ ^∗ = 0.29, N ₂ ^∗ = 0.20, N ₃ ^∗ = 0.28, and N ₄ ^∗ = 0.23 at steady state. The results are coincided with the minimum point of the proposed loss model by the concave optimization theory. With the help of N _Qi = 0.25, the optimal active power allocation coefficients are obtained as N ^∗ _P1 = 0.29, N ^∗ _P2 = 0.19, N ^∗ _P3 = 0.29, and N ^∗ _P4 = 0.23. Then, the output powers from DEUs are distributed as 2.8, 1.85, 2.83, and 2.17 kW by the designed P-E and Q-ω droop controls. By using the aforementioned parameters, the coefficient updating process of the DAOM is smooth, with the well regulation of output currents. The corresponding wire loss, converter loss, and distribution loss by the designed control are 306 W, 680 W, and 986 W, respectively. By comparing with the conventional method, the wire loss is increased by 21 W, while the converter loss is decreased by 132 W. Therefore, the total distribution power loss of the proposed control is reduced by 111 W (reduced by 10.12%). Furthermore, the output powers and bus voltages of DEUs are controlled within the limitations during the whole control process. The final convergence values of current coefficients and active power coefficients are presented in Table 4. These values are consistent with the theoretical minimum point of the loss model. Therefore, both DAOM P-E droop control and Q-ω droop control are proven.

In case 2, the study on the variation of wire resistance is carried out: R₃ is revised as 0.214 Ω (Pecht et al., 2017; Zhang et al., 2019; Singh et al., 2020). Figure 9 shows the control results of the voltages, output currents, current allocation coefficients, active powers, active power allocation coefficients, reactive powers, reactive power allocation coefficients for the DEUs, and the loss. At the initial condition, the current coefficients of DEUs are N ₁ = 0.30, N ₂ = 0.22, N ₃ = 0.22, and N ₄ = 0.26 and the output active power allocation coefficients of DEUs are N _P1 = 0.30, N _P2 = 0.21, N _P3 = 0.22, and N _P4 = 0.26. The output powers are P ₁ = 2.9 kW, P ₂ = 2.05 kW, P ₃ = 2.05 kW, and P ₄ = 2.65 kW and Q ₁ = Q ₂ = Q ₃ = Q ₄ = 0.94 kVar. These output power and current allocation rates are determined by the different wire impedances and voltage for DEUs. Meanwhile, b using the converter loss coefficients and wire resistances in Tables 2 and 3, the loss is calculated as 999 W (the converter loss is 748 W, and the wire loss is 251 W).

When the proposed DAOM is applied at 3 s, the current distribution coefficients, active power allocation coefficients, and reactive power allocation coefficients are continuously updated in a cycle of 1 s. At the same time, the distributed secondary control regulates the output active powers and reactive powers according to the updating power ratio. After iteration, the current distribution coefficients are optimized at N ₁ ^∗ = 0.25, N ₂ ^∗ = 0.25 N ₃ ^∗ = 0.30 and N ₄ ^∗ = 0.20 at steady state. The results are coincided with the minimum point of the proposed loss model by the concave optimization theory. With the help of N _Qi = 0.25, the optimal active power allocation coefficients are obtained as N ^∗ _P1 = 0.250, N ^∗ _P2 = 0.255, N ^∗ _P3 = 0.305, and N ^∗ _P4 = 0.190. Then, the output powers from DEUs are distributed as 2.40, 2.46, 2.94, and 1.85 kW by the designed P-E and Q-ω droop control. By using the aforementioned parameters, the coefficient updating process of the DAOM is smooth, with the well regulation of output currents. The corresponding wire loss, converter loss, and distribution loss by designed control are 271, 628, and 899 W, respectively. By comparing with the conventional method, the wire loss is increased by 20 W, while the converter loss is decreased by 120 W. Therefore, the total loss of the proposed control is reduced by 100 W (reduced by 10.01%). Moreover, the bus voltages and output powers of DEUs in Figure 9 are regulated within the limitations. The iteration results by the DAOM of case 2 are given in Table 5. The optimal values of the current and active power allocation coefficients coincide with the theoretical analysis.

Figure 10 shows the loss comparison. Compared with the traditional method, this control scheme reduces the power loss of the distribution network by about 10.12 and 10.01% in case 1 and case 2, respectively.