Power flow calculation based on local controller impedance features for the AC microgrid with distributed generations

: In AC microgrid, since the impedance features of the power electronic converter are not considered, the accuracy of power flow calculation is degraded. Thus, this study, firstly presents a power flow calculation approach based on local controller impedance features for the AC microgrid consisting of numerous distributed generations. Firstly, the source-side equivalent output impedance matrix of the converter is estimated by small-signal perturbation approach and the equivalent transformation among the dq axis, αβ axis and abc axis. Then, the bus–bus line impedance model is proposed by converting the bus line equivalent impedance model. Moreover, the equivalent impedance model is embedded into Jacobian matrix iterative process to improve the accuracy of the power flow calculation approach. Eventually, the power flow calculation based on local controller impedance features is simulated by IEEE 4-bus MATLAB/Simulink test system, PG&E 69-bus test system and IEEE 118-bus test system, indicating that the proposed power flow calculation approach is more accurate.


Introduction
Almost no one has failed to notice the phenomenon that plenty of distributed generations (DGs) have been applied to modern power systems [1,2]. This is because most of the DGs are the renewable sources; they reduce the carbon emission as well as running cost. Meanwhile, the AC microgrid is a perfect solution to integrate these renewable sources [3]. At present, the major issues related to the AC microgrid are designing of the efficient converter, system voltage/frequency control, energy management and so on [4,5]. Additionally, the multi-objective optimal power management of microgrids has been widely researched [6][7][8]. There into, the sensitivity analysis was proposed to obtain the most optimal location for placement and their optimal control variable setting [7]. The power flow calculation of the power system with numerous DGs is, however, an important issue [9].
Power flow calculation based on Newton-iterative method of the microgrid consisting of various DGs was widely researched in existing the literature [10][11][12][13][14][15][16][17][18][19][20]. From the view point of the distributed sources power fluctuation, the constant model under the maximum power point tracking (MPPT) was embedded into power flow calculation to deal with wind power fluctuation [10]. This approach was, however, ignorance of the lower/upper limits to the rotor speed of the wind turbines. Thus, the power flow calculation model for the doubly-fed induction generators (DIGs) under power regulation was proposed in the literature [11]. Further, considering the variations of both load and distributed sources power, the highfrequency power flow calculation was put forward in time [12]. From the view of the modular multilevel converter (MMC), the mathematical model of the MMC and its equivalent circuit were proposed, and the corresponding power flow calculation model was also provided by the literature [13]. With the development of distributed power systems, the interaction between transmission and distributed power systems could not be neglected. Thus, the global power flow calculation approach where the transmission and distributed power systems were regarded as a whole, was investigated [14]. Recently, some scholars extensively researched the power flow calculation based on modified Newton-iterative method of the AC microgrid dominated by power electronic converter adopting droop control strategy [15][16][17][18][19]. Within the context of the features of the AC microgrid operating in an islanded mode in which the system frequency was no longer fixed, the slack bus could not be applied to power flow calculation. To solve this problem, the conventional Newton Raphson iterative method was improved to compute the power flow for the AC microgrid by a simple and effective modification [15]. What is more, the power flow calculation for AC/DC microgrids considering virtual impedance was studied [16]. Meanwhile, the modified forward-backward sweep method for radial networks and the current injection method for meshed networks were proposed by simulating local generation droop controllers [17]. Then, the power flow calculation method was extended into the droop and isochronously controlled AC microgrids [18]. The main feature of the aforementioned methods was that variable system frequency and droop relationships were incorporated. They are, nevertheless, implemented in the stationary reference frame, and did not provide the essential information to linearise a dynamic model. As a result, the method of determining the operating points of droop-controlled AC microgrids was obtained [19]. In a real power system, the voltage-source converter controlled by active power and reactive power (P&Q) controller still occupied a dominant position [20]. Thus, this paper paid more attention to the power flow calculation of the voltage-source converter-dominated AC microgrids.
The equivalent impedance regarding local controllers of the converters in AC microgrids would impact the line impedance and admittance matrix in Jacobian matrix iterative process. If the local controller equivalent impedance is incorrectly ignored, it would result in an inaccurate or erroneous power flow calculation result. Therefore, the impedance specifications should be considered to reflect the impact of the power electronic converter controllers. Two main types of impedance specification approaches were investigated, i.e. measure-based approach and modelling-based approach [21][22][23][24][25]. The measure-based impedance specifications approach was first proposed for DC microgrids or distributed power systems in literature [21,22]. Furthermore, the measure-or modelling-based impedance specifications approach of the DFIG was first proposed in view of wind farm features to assess the system stability [23,24]. Moreover, the modelling-based equivalent p − n sequence impedance matrix of the converters was studied in [25]. Nevertheless, as far as authors know, the equivalent impedance matrix of the P&Q controller which is obtained by impedance specifications method at the specific frequency (50 Hz) was different from phase reactor impedance, was not studied from the view point of the power flow calculation. There into, the IET Energy Syst. Integr. This is an open access article published by the IET and Tianjin University under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/) 1 equivalent line impedance is the impedance between the converter and the grid, which consists of the impedance of the P&Q controller, filtering impedance and line impedance. As the improvement of renewable energy utilisation, the small scale distributed power generation systems such as AC microgrids have been attractive architectures for a future power system. These small autonomous power systems integrating various DGs and local loads improve the reliability and efficiency of electricity services [26]. Thus, the power flow calculation for such power system needs proposing urgently to satisfy the demand of the economic dispatch, situation awareness and so on [27]. As a result, embedding the features of the DGs controllers, this paper presents a power flow calculation approach for the AC microgrid consisting of plenty of DGs to improve the accuracy of the conventional Newton-iterative power flow calculation. The main features and benefits of this paper are listed as follows: (1) The abc axis equivalent impedance modelling based on impedance specifications in the steady state is built to reflect the effects of the local power electronic converter controller on the system admittance matrix.
(2) The bus-bus line impedance transformation is presented to improve the accuracy of the admittance matrix in the Jacobian matrix iterative process because the output equivalent impedance modelling of DGs is not directly applied to conventional power flow calculation.
(3) The Jacobian matrix iterative process embedding the equivalent impedance modelling is proposed to improve the accuracy of the power flow calculation approach.
The rest of this paper is arranged as follows. This paper establishes the abc axis equivalent impedance modelling matrix based on impedance specifications of the DGs adopting P&Q control strategy in Section 2. In Section 3, the bus-bus line equivalent impedance transformation approach and Jacobian matrix iterative process embedding the equivalent impedance modelling are proposed. In Section 4, the proposed power flow calculation is tested via comprehensive simulation results of the IEEE 4-bus MATLAB/Simulink test system, PG&E 69-bus test system and IEEE 118-bus test system. Eventually, the conclusion is provided in Section 5.

ABC axis impedance modelling matrix of the voltage-source converter
This section firstly illustrates the typical controller structure figure of the voltage-source converter shown in Fig. 1 [28] (Fig. 3 in [28]). Further, the source-side output impedance matrix is built to reflect the impact of the local controller on the admittance matrix. If the DG converter controlled P&Q controller provides the constant active and reactive powers at the PCC, the converter impedance can be ignored/bypassed by the P&Q references entered to the DG converters. Also this impedance can be neglected.
However, in practical microgrid, the rated active power and reactive power (P ref and Q ref ) in Fig. 1 are decided by the maximum output power of distributed renewable sources controlled by MPPT. Thus, the impedance of the P&Q controller, filtering impedance and line impedance should be regarded as the whole equivalent line impedance from the distributed renewable source output port to PCC. A similar concept can be found in the literature [16], where virtual impedance appeared by the virtual impedance controller is embedded to power flow calculation. Thus, this paper pays more attention to the condition that the P&Q references are decided by the maximum output power of distributed renewable sources controlled by MPPT.
Meanwhile, the current/voltage dynamic modelling of the P&Q controlled converter is written as follows: where V d , V q , I d and I q are voltage-source converter output voltages and currents in the dq axis, respectively. V 0d , V 0q , I 0d and I 0q are voltage-source converter voltages and currents in the dq axis, respectively. C line , L line and R line are the line output capacitance, line output inductance and line output resistance, respectively. ω is the grid angular speed, and s is the Laplace operator. Moreover, the inner voltage/current loop PI controller is embedded into the controller to improve the performance of the output voltage/current which is shown as follows: where I d * , I q * , V d * and V q * are the voltage-source converter current and voltage signals in the dq axis, respectively. G v and G i are voltage/ current double loop PI controller (G v = k vp + k vi /s and G i = k ip + k ii /s), and K is a feed-forward gain. Moreover, the instantaneous apparent power is obtained by the voltage-source converter output current and voltage in the dq axis which is shown in (3). In order to measure the output active and reactive powers without possible oscillations, the low-pass filter is applied to obtain the averaged active/reactive powers as shown in (4) where ω c is low-pass filter cutoff frequency. For small perturbations around the state-steady point, the linearised equations can be applied to source-side impedance modelling of the voltagesource converter in the dq axis, which is represented as follows: where ω * is the main grid angular speed, and we can approximate ω * = 100 πrad/s in practice.
According to (1)-(5), the impedance modelling matrix of the voltage-source converter in the dq axis (Z dq ) has been obtained by (5). However, the impedance modelling matrix in the dq axis is not directly applied to power flow calculation, and it must be converted into the impedance modelling matrix in the abc axis. In light of Z 1 and Z 2 , the impedance modelling matrix of the voltage-source converter in the dq axis is a symmetric component. As a result, the sequence-domain impedance matrix could be provided through the relationship between the dq frame-domain and the p − n framedomain [29]. Thus, the positive-negative sequence impedance matrix in the pn axis is presented in (6) Since the coupling sequence components are zero on account of the definition of the mirror frequency decoupled system (Z pn = 0 and Z np = 0) [30], (6) can be switched to (7) further where Z pn , diagonal matrix, is the p − n sequence impedance matrix. Z pp and Z nn are pp and nn sequence impedance values, respectively. Moreover, the phasors impedance matrix in the abc axis can be provided from the sequence-domain impedance matrix in the pn axis, which is shown as follows: where a = e j(2π/3) is the complex number which corresponds to a 120° phase shift. Z 00 is 00 sequence impedance, which is neglected in this condition [29] (Z 00 = 0). Therefore, the equivalent impedance matrix of the voltage-source converter in the abc axis can be represented in Fig. 2.

Modified power flow calculation based on local controller impedance features
In this section, the modified power flow calculation based on the above equivalent impedance matrix model of the voltage-source converter in the abc axis is proposed. As shown in Fig. 3, the equivalent output impedance model of the voltage-source converter in the abc axis needs to be converted into the bus-bus line impedance, which is the impedance from bus i to bus j and is directly applied to system admittance matrix in Jacobian matrix iterative process. Suppose that the output equivalent impedance of the voltagesource converter is Z abc . Meanwhile, the DG is link to the bus 0 which is connected to bus 1, …, n. According to Kirchhoff's current Law, the output current of the voltage-source converter is as follows: where n represents the number of the buses connected to bus 0. As shown in Fig. 4, where V 0 = V 1 ′ = ⋯ = V n ′, the (9) can be rewritten as follows: where can be given as follows: Further, (12) can be provided by (11), which is shown as follows: Thus, (13) and (14) can be obtained to provide the equivalent impedance of the local controller.
where j = 1, 2, …, n. S is the apparent power. Thus, if the DG is installed in the bus i, the line impedance from bus i to j can be obtained as follows: where Z abci ↔ j is the line impedance from bus i to j without considering the local controller of the DG i , and Z abci ↔ j * is the whole line impedance from bus i to j with the local controller equivalent model. Thus, the nodal impedance matrix Z B and nodal admittance matrix Y B can be obtained, where Y B = [G i, j + jB i, j ] m * m and m is the number of the buses. According to the literature [6], the active/ reactive powers' imbalance injection and voltage imbalance injection can be shown as follows: where e di and e qi are the real part and imaginary part of the ith bus voltage, respectively. Thus, the modified power flow calculation process is shown in Fig. 5.

Simulation
In this section, the power flow calculation is tested in MATLAB2014a, utilising a personal computer with intel(R) Core(TM) i5-4590 CPU @ 3.30 GHz and 4 GB of RAM. The IEEE 4-bus MATLAB/Simulink test system is provided to illustrate the accuracy of the proposed power flow calculation. Furthermore, the radiation network power system and loop network power system are separately tested to verify the performance of the proposed modified power flow calculation with different penetration of the distributed renewable energy, which is shown in Figs. 6 and 7. The detailed system parameters are taken from the website clicked on https://figshare.com/articles/ IEEE_system_impedance/7967297.

IEEE 4-bus simulation test system
In this section, the IEEE 4-bus MATLAB/Simulink test system is provided to illustrate the accuracy of the proposed power flow calculation. Meanwhile, the parameters in the local controllers of the voltage-source converters are presented as follows: G i = 10/s + 800, G v = 0.9/s + 1.5, L line = 2 mF, R line = 0.00 5Ω, C line = 5 μF and ω * = 100 πrad/s, ω c = 20 πrad/s, K = 0.1. As shown in Fig. 8, the DG is installed in the third bus of the IEEE 4-bus MATLAB/ Simulink test system. As shown in Figs (19) and (20). where

Radiation network power system
In this subsection, the radiation network power system (PG&E 69bus) is tested to reflect the impact of the local controller of the voltage-source converter on power flow calculation. In order to simplify the calculation, the parameters of the local controllers of the voltage-source converters are the same, which are shown as follows: G i = 10.05/s + 251.3, G v = 0.265/s + 10, L line = 2 mF, R line = 0.15 Ω, C line = 45 μF and K = 0.3, ω * = 100 πrad/s, ω c = 20 πrad/s. As shown in Fig. 6, the DGs are installed in the 12th bus, 26th bus, 58th bus and 66th bus of the PG&E 69-bus radiation network power system. As shown in Figs. 11 and 12, the red line represents the proposed power flow calculation results, and the blue line represents the conventional power flow calculation results. Meanwhile, the bus voltage magnitude and angle are separately obtained, which illustrates that the power flow calculation results will be changed via considering the local controller of the voltage-source converters. From the viewpoint of the proposed method, only the partial admittance matrix is changed due to the DGs controlled by P&Q controller. Furthermore, a large proportion of buses are the traditional PQ or PV buses, which does not result in the change of the admittance matrix. Thus, the voltage magnitudes of only some bus groups are differently obtained in different methods. Furthermore, the voltage magnitude and voltage angle errors can be obtained by (21) and (22)

Loop network power system
In this subsection, the loop network power system (IEEE 118-bus) is tested to reflect the impact of the local controller of the voltagesource converter on power flow calculation. Similar with the above subsection, the parameters in the local controllers of the voltagesource converters are also the same, which are shown as follows: G i = 4/s + 800, G v = 1/s + 3, L line = 2 mF, R line = 0.05 Ω, C line = 5 μF and ω * = 100 πrad/s, ω c = 20 πrad/s, K = 0.3. It is shown in Fig. 7 that the DGs are installed in the 2nd bus, 20th bus, 33th bus and 114th bus of the IEEE 118-bus loop network power system. As shown in Figs. 15 and 16, the red line represents the proposed power flow calculation results, and the blue line represents the conventional power flow calculation results. The power flow calculation results with/without considering the impact of the local controller are shown in Figs. 15 and 16, which explains that the power flow calculation results will be changed via considering the local controller of the voltage-source converters. According to (21) and (22), the voltage magnitude and voltage angle errors between proposed power flow calculation results and conventional power flow calculation results are E Vr1 = 0.366% and E δr1 = 1.158%, respectively. Moreover, the total loss of power is also changed. In addition, the DGs are continued to be added in the 52th bus 67th bus, 88th bus and 97th bus of the IEEE 118-bus radiation network power system. As shown in Figs. 17 and 18, the bus voltage magnitude and angle are separately obtained, which illustrates that the power flow calculation results will be greatly changed by considering the local controller of the voltage-source converters. Meanwhile, the voltage magnitude and voltage angle errors between proposed power flow calculation results and conventional power flow calculation results are E Vr2 = 3.58% and E δr2 = 8.06%, respectively. As a result, with the penetration of distributed renewable energy increasing, there is no doubt that the local controller of the voltage-source converter is more suitable to be embedded into power flow calculation similar with the results of the radiation network power system. Eventually, the computation time of the proposed power flow calculation method and the conventional power flow calculation method are 0.176 and 0.075 s, respectively. As a result, the computational burden of the proposed power flow calculation method will be increased.

Conclusion
This paper has presented a modified power flow calculation approach based on local controller impedance features for the AC microgrid consisting of numerous DGs to satisfy the power flow calculation accuracy demand of the microgrid economic dispatch, situation awareness and so on. Since the impedance matrix of the local controller in the voltage-source converter impacts the admittance matrix in power flow calculation process, the impedance specifications model in the dq axis has been proposed to reflect this impact. According to frame-domain transformation relationship and symmetric component feature, the impedance matrix in the dq axis has been switched into the impedance matrix in the abc axis. As the output equivalent impedance model of DGs is not directly applied to power flow calculation, the bus-bus line impedance transformation has been proposed to improve the accuracy of the Jacobian matrix. In the end, the theoretical power flow calculation analysis results has been verified by the IEEE 4bus MATLAB/Simulink test system, PG&E 69-bus test system and IEEE 118-bus test system. In the future, the more universal converter which exists the mirror frequency coupled modes should be further researched to improve the accuracy of the conventional power flow calculation. Moreover, the power flow calculation based on local controller impedance features for the islanded microgrid dominated by droop controlled or virtual synchronous generator controlled converter should be studied.