The equivalent circuit of the hybrid connected inverters is shown in Figure 2. ICI-based and CCI-based DG units are coupled to the power grid with different DC sources. The general power flow from the DG unit to the power grid can be expressed as: P = V V p c c Z cos θ − δ − V p c c 2 Z cos ⁡ θ Q = V V p c c Z sin θ − δ − V p c c 2 Z sin ⁡ θ (1) where V p c c and V are the magnitudes of the voltage of the power grid and inverter, and δ is the phase difference between the voltage of the inverter and the power grid. Z and θ are the magnitude and phase angle of the coupled impedance. With different coupling structures for ICI and CCI, Z and θ are also different as shown in Eqs 2, 3:   Z L = ω L θ L = 90 ° (2)   Z C = 1 ω C C = 1 ω C − ω L C θ C = − 90 ° (3)

To generalize the power flow, a power base is defined as: S b a s e − L = V p c c 2 Z L S b a s e − C = V p c c 2 Z C (4)

By standardizing all ICIs as one ICI unit and all CCIs as one CCI unit, the power output of the inverters in Figure 2 can be expressed as: P L Q L P C Q C = sin ⁡ δ V p c c S b a s e − L cos ⁡ δ V p c c S b a s e − L 0 0 0 0 sin ⁡ δ V p c c S b a s e − C − cos ⁡ δ V p c c S b a s e − C V L V C + 0 − S b a s e − L 0 S b a s e − C (5)

Usually, there are three factors that affect the power control range of the inverter: the property of the coupling impedance (inductive or capacitive), the output voltage, and the equivalent value of the coupling impedance. The following discussion focuses on these three aspects in describing the advantages of the proposed MG system. Since the output voltage is proportional to the DC-link voltage, these two voltages are not distinguished in the following discussion.

The main difference between ICI and CCI is the property of the coupling impedance, which leads to different power delivery characteristics. Assuming the direction of active power injection and reactive power compensation is positive, the power base and the DC-link voltage of the ICI and CCI are the same, and the last term is > 2 V p c c . Thus, the power range boundary of the ICI and CCI can be illustrated as a circle, in which the points are at the power base and the radii of ICI and CCI are V L V p c c / Z L and V C V p c c / Z C , where V L and V C are the output voltage of the ICI and CCI, respectively. Figure 3A depicts the positions of the power range circles of the ICI and CCI, and the total power range of the hybrid connected CCI and ICI. The total power ranges of the traditional parallel-connected ICIs are shown in Figure 3B.

In Figure 3, P L − max and P C − max are the maximum active power transfer boundaries of ICI and CCI, respectively; Q L and Q C indicate the reactive power compensation range of these two inverters, respectively; P H − max and P H − max are the maximum active power transfer boundaries of the proposed hybrid connected system, respectively; and Q H is the power compensation range of a parallel-connected ICI system.

According to Figure 3, it can be found, that with the proposed system, the active power transfer range of the two MG systems is the same, and the reactive power compensation range of the proposed system is much larger than the traditional one.

The output voltage of the inverter will also affect the power transfer. Figure 4 shows the power range with the variation of the output voltage of ICI and CCI.

The CCI achieved a wide range of reactive power compensation with lower output voltage, and the DC-link voltage requirement of CCI could be effectively reduced, which increased the adaptiveness and flexibility of the MG system in reactive power compensation (Figure 4).

The equivalent value concept of the coupling impedance can be replaced by the power base. Practically, the power base of the CCI is usually set equal to the nominal load demand of the MG. In this paper, the CCI’s power base is set as a reference like the following: S b a s e = S b a s e − C The ratio of the ICI power base to the reference power base was set as: λ = S b a s e − L S b a s e (6)

If the battery charging in a MG is not considered, the power range boundaries of the proposed MG system can be deduced (Table 2).

To sum up, we believe that the proposed hybrid of ICIs and CCIs connected to a MG system is able to achieve a wider reactive power compensation range. Since the CCI is able to work under a lower DC-link voltage, an optimal power sharing method could reduce the total power capacity compared with a traditional microgrid using ICIs only.

The power-sharing ratio was defined for the proposed MG system. The power reference of the hybrid-connected ICI and CCI can be described as: P L = m * P o u t (7) Q L = n * Q o u t P C = 1 − m * P o u t Q C = 1 − n * Q o u t

where P o u t and Q o u t represent the total power injected into the PCC, P L and Q L are the power references of the ICI, and P C and Q C are the power references of the CCI.

To normalize the power demands, power reference coefficients are defined as: P o u t Q o u t = λ S b a s e r p r q (8)

The power capacity of ICI and CCI can be deduced as: S L = V L i L S C = V C i C (9)

where i L and i C are the output currents from ICI and CCI, respectively. They are expressed as: i L = P L 2 + Q L 2 V p c c = m P o u t 2 + n Q o u t 2 V p c c i C = P C 2 + Q C 2 V p c c = 1 − m P o u t 2 + 1 − n Q o u t 2 V p c c (10)

According to Eq. 9 and Eq. 10, the power capacity of an inverter changes with the power ratio for a given set of active and reactive power references. A detailed discussion will be provided hereinafter.

To better evaluate the effect of power sharing ratio on power capacity, a case study is essential.

Assuming that the power base of the ICI was 1/8 times that of the CCI and the ratio of DC-link voltage to PCC voltage for the ICI was 1.5 and for the CCI, 0.7, six cases with random power reference coefficients are listed in Table 3.

The positions of the case studies in terms of the power delivery range of the CCI are shown in Figure 5. As shown in the figure, the active power reference in case 1 was much larger than the reactive power reference. In cases 2, 3, and 4, there was little difference between the active and reactive power references, which were uniformly distributed outside the CCI’s power output range. The power references in case 5 both fell within CCI’s power range. To keep the proposed system always working within the minimum power capacity, after receiving the power demand signals, the current from the CCI should be as large as possible because of the lower DC-link voltage, and to keep minimum power capacity the current from the CCI must obey the relationship: i C i L ≤ V L V C

By substituting the power reference coefficients into (Eq. 11), we noted that the current from the ICI varied according to the different power-sharing ratios (Figure 6). Thus, the power sharing ratios significantly affected the output current.

According to Figure 6, the lowest point is always located at (0, 0) in the x-y plane, which means the ICI should never output any current on mathematical calculation. However, due to the limitation of voltage and the power base, a single CCI may not always satisfy the power demands. Therefore, to find the correct power sharing ratio to meet the power demands and reduce the ICI current to the lowest possible, a power sharing ratio calculation method is proposed in the next part.

Power capacity is seriously restricted by the inverter’s output voltage and PCC voltage. Eq. 11 described the relationship between voltage and power, as well as power references as: 1 + Q L λ S b a s e 2 + P L λ S b a s e 2 ≤ V D C − L 2 V p c c 1 − Q C S b a s e 2 + P C S b a s e 2 ≤ V D C − C 2 V p c c (11)

The power sharing ratio boundary is also obtained as: 1 + n r q 2 + m r p 2 ≤ V D C − L 2 V p c c 1 − 1 − n r q 2 + 1 − m r p 2 ≤ V D C − C 2 V p c c (12)

In the hybrid MG system, the output current of the ICI should be as small as possible because of its higher DC-link voltage. This calculation is expressed as: S L + S C < S e q u a l (13)

Since only the CCI is active in case 5, power sharing is not available. The current vector graphs for power-sharing are illustrated in Figure 7 for cases 1–4.

In the figure, i L − e is the output current from the ICI under equal power sharing, and it can be obtained as: i L − e = r p 2 + r q 2 2 V L S b a s e (14)

For this case study, the number of ICIs under equal power sharing was assumed to be 2.

In Figure 7, i p − r e f , i q − r e f are the active and reactive power vectors in the reference current, respectively. They can be obtained as: i p − r e f = r p V p c c S b a s e i q − r e f = r q V p c c S b a s e (15) where i q C is the reactive power vector current when the CCI compensates for the reactive power equal with the power base as: i q C = S b a s e V p c c (16)

The radius of the circle of the CCI’s power range can be described as: r c = V C V p c c 2 ∗ S b a s e (17)

The initial point and the location of the power reference are marked as α 1 and α 3 , respectively. Their positions on the coordinate are: α 1 → = 0 + j 0 α 3 → = i q − r e f + j i p − r e f and i C and i L can be calculated as: i L = α 3 → − α 2 → i C = α 2 → − α 1 →

Assuming the acute angle between r c and the horizon axis is β , the point α 2 gives the maximum current from the CCI with a given power reference. It has three positions:

1. When r q > 1 , the current is:

The power-sharing ratio is: m = i p − r e f − r c 1 − 1 1 + tan 2 ⁡ β V L r p S b a s e n = i q − r e f − r c 1 1 + tan 2 ⁡ β + i q C V L r q S b a s e (21) and: tan ⁡ β = i p − r e f i q − r e f − i q C (22)

2. When r q < 1 , the current is:

The power-sharing ratio is: m = i p − r e f − r c 1 − 1 1 + tan 2 ⁡ β V L r p S b a s e n = i q − r e f − i q C − r c 1 1 + tan 2 ⁡ β V L r q S b a s e (24) and: tan ⁡ β = i q C − i q − r e f i p − r e f (25)

3, When r q = 1 , β = 90 ° , and r p is larger than the maximum active power transfer of CCI, the current is:

According to Eqs 14–30, the power sharing ratios can be obtained for cases 1–4. The power sharing ratio in case 5 is determined as (0, 0). Thus, the corresponding power ratios under the proposed method and the conventional method were calculated and are summarized in Table 4. As the comparison shows, the proposed power sharing method for the hybrid MG system can effectively reduce the power capacity.

The architecture of an NN model is based on neurons and can be divided into three main parts: the input layer, the hidden layer, and the output layer (Wang et al., 2020). The input layer receives signals from the outside world. The hidden layer is the intermediate layer connecting the input layer and the output layer, and constitutes the core of the NN as being composed of many neurons. The output layer returns the results of the mathematical operations on the input (Liu et al., 2021).

As a representative of a NN the application areas of a back-propagation neural network (BPNN) include function approximation, regression analysis, numerical prediction, and classification and data processing (Chen et al., 2020). The BPNN calculates the final network error based on the forward operation of the input data, transmits the error in the opposite direction, and contains a mechanism for adjusting the weights and thresholds of the corresponding layers according to certain rules when crossing different layers. After a large number of data samples are trained, an algorithmic model that can accomplish complex nonlinear mapping is finally constructed. For such networks, the input layer is responsible for feeding data into the neural network. The hidden layer fits and optimizes the parameters to the neural network by model training. These include weights and biases as well as the selection of various parameters and activation functions. The output layer is responsible for outputting the computational results from the neural network (Wang et al., 2022). The principal algorithm process of the BPNN model in this paper is illustrated in Figure 8.

The BPNN model can be expressed mathematically as f ∑ i X i W i + b i , where f is the activation function, which introduces non-linear factors to a linear model, thus solving problems that are difficult to solve with a linear model. X i represents the output data, W i is the weight between the connected input data and the hidden layer, while b i is the bias vector. During the training process, the accuracy of the NN model is improved by continuously changing the values of the parameters W i and b i .

A loss function was set to examine the difference between the predicted value and the known answer, and also to evaluate the accuracy of the trained NN model. With enough training sessions, the loss function will be stable in the vicinity of a very small value, which can be regarded as the prediction error of the NN model (Huang et al., 2021). In this paper, the sigmoid function was chosen as the activation function and the mean square error (MSE) as the loss function to describe the prediction error.

For a BPNN, the number of neurons, hidden layers, as well as training sessions, have a high positive correlation with the final MSE. However, an excessive number of neurons and hidden layers will increase the computational power required of the chip in the power sharing controller. In addition, an increase in the amount of training data will also increase the difficulty of BPNN training (Huang et al., 2021). Therefore, it is essential to choose the proper numbers of neurons and hidden layers, and also the amount of training data. The selection principle is based on experience, as well as the required prediction error of the BPNN.

To evaluate the impact on the total power capacity due to the prediction error, a partial derivative equation is presented. Assuming that the prediction error of the total power capacity is defined as: ∆ s = ∂ S ∂ m ∆ m + ∂ S ∂ n ∆ n (31)

there are: ∂ S ∂ m = ∂ S L ∂ m + ∂ S C ∂ m (32) ∂ S L ∂ m = 1 2 2 m r p 2 1 + n r q 2 + m r p 2 n r q 2 + m r p 2 S b a s e − L + 1 + n r q 2 + m r p 2 2 m r p n r q 2 + m r p 2 1 2 S b a s e − L (33) ∂ S C ∂ m = m − 1 r p 2 S b a s e − C × 1 − m r p 2 + 1 − n r q 2 1 − 1 − n r q 2 + 1 − m r p 2 + 1 − 1 − n r q 2 + 1 − m r p 2 1 − m r p 2 + 1 − n r q 2 (34) and: ∂ S ∂ n = ∂ S L ∂ n + ∂ S C ∂ n (35) ∂ S L ∂ n = 1 + n r q r q S b a s e − C × m r p 2 + n r q 2 1 + n r q 2 + m r p 2 + 1 + n r q 2 + m r p 2 m r p 2 + n r q 2 (36) ∂ S C ∂ n = 1 − 1 − n r q r q S b a s e − C × 1 − m r p 2 + 1 − n r q 2 1 − 1 − n r q 2 + 1 − m r p 2 + 1 − 1 − n r q 2 + 1 − m r p 2 1 − m r p 2 + 1 − n r q 2 (37)

The results from inserting the parameters from cases 1 to 5 into (Eq. 26) are summarized in Table 5.

As in Table 4, the degrees of impact of m and n on the total power capacity are not the same. As the reactive power sharing ratio will more critically affect the prediction error, a larger active power demand will increase the effect of the reactive power sharing ratio error on the power capacity. A smaller reactive power output will reduce the influence of the active power sharing ratio on the power capacity.

Assuming that the total power capacity bias should not exceed 5%, the prediction errors of the active and reactive power sharing ratio must be smaller than 0.5%, which means that the MSE of the BPNN cannot be larger than 0.005 to keep the total power bias within a satisfactory range.

The BPNN model includes two hidden layers, and a total of 15 neurons were proposed. The size of the training set was 380 groups of power references and sharing ratios. Like the training results shown in Figure 9, after training, the MSE of the BPNN model reached a value on the order of 10^–5, which is satisfactory for generating power sharing ratios.

The control block diagram of the proposed power sharing method for the hybrid ICI and CCI system microgrid is presented in Figure 10. As the figure shows, the power demands will be sent to the secondary power sharing layer with a well-trained BPNN model and generate an optimal power sharing ratio, which can reduce the total power capacity requirement of the hybrid MG system to the minimum value. The power sharing ratio then will be used to calculate the actual power references for the local inverters. Virtual-impedance loops are employed in the local control layers to eliminate the tracking errors by droop control.

Simulation tests were conducted in Matlab/Simulink, to verify the effectiveness of the proposed power sharing strategy. Test cases are listed in Table 3, and the control block diagram is given in Figure 10. Simulation parameters are listed in Table 6. All tests were performed with ICIs under equal power sharing and using the proposed adaptive power sharing model. The simulation results with the proposed MG system and adaptive power sharing method are illustrated in Figure 11 and the results of equal power sharing with the ICIs in the MG system are shown in Figure 12. Power references for each inverter were calculated based on the total power references and power sharing ratio; the references were varied by 1.5 s in each case. Based on the results, the power sharing algorithm proved effective and the output power tracked the reference smoothly and accurately.

To verify the power control effectiveness of the proposed method on the hardware level, a prototype was implemented in the lab (Figure 13). The configuration of the testing system was the same as the simulation. The parallel-connected inverters were controlled by a DSP-based controller. All tests followed the simulation settings. The power transfer waveforms under the different cases are shown in Figure 14.

Similarly, the conventional equal power sharing method was also tested on parallel-connected ICIs. The power transfer waveforms are provided in Figure 15.