New adaptive droop control with combined line impedance estimation method for parallel inverters

This paper presents a new load sharing control between paralleled three-phase inverters in an islanded-microgrid based on the line impedance estimation online by the use of the Kalman filter. We can solve the mismatch of power sharing when the line impedance changes due to the temperature and frequency, significant differences of line parameters and requirements of Plug-and-Play mode of inverters connected to the microgrid. Moreover, the paper also presents a new Droop control method working with the line impedance which is different from the Droop traditional algorithm when the line impedance is assumed pure resistance R or pure inductance X. In the paper, the line impedance estimation for parallel inverters uses the least squares method combined with Kalman filter. In addition, secondary control loops are designed to restore the voltage amplitude and frequency of the microgrid by using a combined nominal value SOGI-PLL with generalized integral block and phase lock loop to exactly monitor the voltage magnitude and frequency phase at common PCC. Control model has been simulated in Matlab/Simulink with three voltage source inverters connected in parallel for different ratios of the power sharing. The simulation results have shown the accuracy of the proposed control method. Therefore, the proposed adaptive droop control method based on line impedance estimation can be an alternative one for load sharing control in islanded microgrids.


INTRODUCTION
With the expansion of the electrical power grid, the conventional power system has become increasingly vulnerable to cope with the reliability requirements and the diverse demand of power users.Moreover, distributed generation (DG) has appeared to advantages such as pollution reduction, high-energy utilization rate, flexible installation location, and DEVELOPMENT, Vol 19, No.K7-2016 Trang 46 low-power transmission losses [1]- [2].DG units have also a higher degree of controllability and operability compared to the conventional generators which will allow microgrids to play a major and critical role in maintaining the reliability and stability of electric networks [3]- [6].Therefore, microgrids will gradually become a strong and effective support for the main power grid and a potential one for the future trends of power systems [7].
In fact, the renewable energy resources such as the wind, solar and tidal energy are connected to the conventional grid through the converter today and the microgrids are formed before they are connected to the grid [8]- [12].In the grid-connected mode, the DG units are often controlled as grid-following.The most adopted control strategies for grid-following inverters are discussed in [4], [7], [13]- [14].When a microgrid is operating in the islanded mode, each DG unit should be able to supply its share of the total load in proportion to its rating.The control strategies for this mode are usually divided into two main types [11], [15] as follows.The first type is communication-based control techniques including concentrated control, master/slave control, and distributed control.These techniques can achieve an excellent voltage regulation and proper power sharing.However, these control strategies which require communication lines between the modules may result in the increased cost of the system.Long distance communication lines will be easier to get interfered, thus reducing the system reliability and expandability.The second type is based on the droop control technique without requiring communications and it is widely used in conventional power systems [2]- [3], [8], [16]-[22].The reason for the popularity of this droop control technique is that it provides a decentralized control capability that does not depend on external communication links in the control strategy.This technique enables the "plug-and-play" interface and enhances the reliability of the system.However, the communication can be used in addition to the droop control method to enhance the system performance without reducing the reliability [23]- [30].
Traditional droop control techniques have some disadvantages such as slow response to changes of load, inaccuracy in power sharing, unbalanced harmonic current, and dependent on the line impedance of inverters [11].In addition, difficulties in the power sharing also are due to the reasons as follows:  The line impedances are not available and different to each others.This affects a lot to The frequency droop is used to control the reactive power sharing and an integral control term is added to the voltage droop to maintain the accuracy of the real power sharing.However, load changes during the compensation period or between the compensation periods may result in a poor power sharing.On the other hand, the analysis and the control strategy introduced in This is based on the assumption that the phase angle difference between the voltages at the PCC and the inverter output is negligible.This assumption may not hold for long feeders or for higher power levels.
This paper proposes a new method of droop control allowing an accurate load sharing ratio between the paralleled inverters in the islanded microgrids with line impedance estimated online in terms of the conventional resistor.Moreover, the line impedance may vary according to the temperature or frequency at the same time with significant differences between the inverters.The estimation blocks provide the line impedance parameters in the real time line for the proposed droop controller which was built based on the least squares method combined with the Kalman filter.In addition, SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 19, No.K7-2016 Trang 48 secondary control loops are designed to restore the voltage amplitude and frequency of the microgrid by using a combined nominal value SOGI-PLL with generalized integral block and phase lock loop to exactly monitor the voltage magnitude and frequency phase at common PCC.Therefore, the proposed adaptive droop control method can be an alternative one for load sharing control in islanded microgrids.

Microgrid Structure in Islanded Mode
The structure of an islanded microgrid composes of many inverters connected in parallel.In Figure 1

ISLANDED MICROGRID CONTROL 3.1. The proposed droop control
The principle of the droop control method is explained by considering an equivalent circuit of an inverter connected to the AC bus.The analysis method is based on the Thevenin theorem as shown in Figure 2. The active and reactive power supplied by the inverter is calculated as follows: In general, both inductance X and resistor R are considered.The use of an orthogonal linear rotational transformation matrix T from the active power P and reactive power Q to the active power P' and reactive power Q' is determined by: When the power angle  is small, equations (1), (2) and ( 3) can be rewritten as: From ( 4), the basis for the well-known frequency and voltage droop regulation through active and reactive power is calculated by: on the parameters of the system.From ( 5) and ( 6), we will have: Combine all equations (1), ( 2), ( 3), ( 5), ( 6), ( 8) and ( 9), we have conditions for the accurately rated power sharing as in (10): To satisfy (10), we must choose the droop coefficients that are proportional to the line impedance.if we adjust the system to meet requirements, the droop will affect the quality of frequency and voltage.Therefore, we have proposed an adaptive controller droop to ensure the accurate power sharing of parallel inverters.

The proposed real power sharing controller
The proposed droop controller still uses the equation in ( 6) and the voltage of the inverter will be calculated as: is the voltage at PCC.


In equation ( 13  is the output of the reactive power sharing controller. Linearize ( 11), ( 12) and ( 13) around , , , , , we will have: The relationship among ( 15), ( 16) and ( 17) is shown in Figure 3.The transfer function of Figure 3 will be as follows: From ( 18), we can calculate: The transfer function (18) has shown that the constant of loops control can be adjusted by 1 p k , not by 1 p m .The real power sharing will not affect the quality of voltage and frequency anymore.

The proposed reactive power sharing controller
The proposed droop controller still uses the equation in ( 5) while the voltage angle of the inverter will be calculated as: Where 1 p k is the gain of the integral and PCC  is the angular frequency at PCC.
In equation ( 14), R1 and X1 are the output from the line impedance estimation,   and   are the output of SOGI-PLL blocks, 1S V is the output of the real power sharing from the controller as mentioned above.
Linearize ( 14), ( 19) and (20) around The relationship among ( 21),( 22) and ( 23) is shown in Figure 4.The transfer function in Figure 4 will be as follows: From (24), we calculate The transfer function ( 24) has shown that the constant of the loops control can be adjusted by 1 p k , not by 1 q m .The real power sharing will not affect the quality of voltage and frequency anymore.
Equations ( 11) and ( 19) have shown that when the system achieved the steady-state, the measured voltage of the inverter will be equal to the rated voltage.The proposed droop control has solved the mismatch of power sharing caused by the different impedances of transmission lines.The rated power is always achieved by the controller.

The line impedance estimation method 3.2.1 The recursive least squares method (LSM)
SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 19, No.K7-2016 Trang 52 The equivalent three-phase circuit of the inverter connected to loads is shown in Figure 5.According to the equivalent circuit in Figure 5, we can write as follows: Equation ( 25) can be rewritten as follows: By discretization of the equation ( 26), we obtain: The transition matrix is described as follows: where T is the sample cycle used to discretize the system.

1
, 27) represents the relationship between the input and output of the object as follows: Object where e(k) is the measurement and process noise.
The relationship between the input and output of (28) can be written as follows: The problem is to estimate the parameters of vector θ based on the current data and voltage.Neglecting the noise e(k), we have predicted the linear regression: The store of all the sample data in the real time and calculation of the volume do not increase much time due to using the recursive least squares method.This algorithm includes the equation as follows: where  is the forget coefficient selected in the range from 0.98 to 0.995.
The line impedance is estimated by a technique based on the recursive least squares method (LSM).The parameter vector LSM  determined from the measured chain value should be affected by the noise or error in equation ( 31).Therefore, we use the Kalman filter to filter out the noise and obtain the value of Kalman  approximate with the real value.

Using the Kalman filter algorithm to filter noise for θ
The Kalman filter is to estimate a process by using a form of the feedback control.The process of the Kalman filter is shown in Figure 6.The Kalman filter firstly estimates the state of the process at a time and then gets the feedback from the measured value to correct the estimation.Therefore, the equation of the Kalman filter is composed of two groups including the time update group and measurement update group.
The equations for measurement updated to correct estimation: . .36 where K is the Kalman gain.

SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 19, No.K7-2016
Trang 54 The start of Kalman filter algorithm is initialized at the initial values: Equations ( 34)-( 36) are applied to the Kalman filter and the procedure is repeated until the difference between the actual value and the value estimated less than a predetermined error ε.The result at the output of the Kalman filter is a vector From (37), we obtain the value of RKalman, LKalman.   Figure 8a shows the frequency response of the SOGI-PLL when the frequency of the input signal changes from 50Hz to 48Hz at t = 0.5s and from 48Hz to 50Hz at t = 1s.Figure 8b shows the frequency response of the SOGI-PLL when the phase angle of the input signal changes from 0 o to 45 o at t = 0.5s. Figure 8c shows the response of the input and output voltages of the SOGI-PLL.The simulation results in Figure 8 have shown that SOGI-PLL can exactly obtain the voltage amplitude and frequency at the point of common coupling (PCC).They will be the input for inner-controller.So when we have more exact values, we will get more accurate power sharing.

SIMULATION RESULTS AND DISCUSSION
A microgrid with two parallel DG units as in Figure 1 is simulated in Matlab/Simulink.All the simulation parameters of the system are given in Table 1.In this simulation, the line parameters of two inverters are provided in Table 2.The results from the simulation are given in Figure 9.When the line parameters change at 3s and 6s, both strategies have normal real power sharing but the conventional droop control can't achieve accuracy reactive power sharing because of the mismatch in line impedances.Only the proposed strategy with line impedance estimation block (Figure 9e, 9f) can share accurate real and reactive power in 1:1 ratio.The voltage drop is always in the limit (Figure 9g).

4.2.
Line parameters and loads change with the same power sharing ratio.
The line parameters of two inverters for this simulation are provided in Table 3.The obtained simulation results are given in Figure 10.In this case, the line parameters change at 3s and 6s, the loads change at 2s and 5s, the power sharing ratio is 1:1.With conventional droop control (Figure 10c, 10d), the real power sharing get worse when the loads and line parameters change, the reactive power can't share at 1:1 ratio if the mismatch is large.Because in the period from 3s to 9s, the actual parameters of the line changes as in Table 3 while the parameter setting of the inverters is unchanged since the conventional controller is not associated with the line impedance estimation for this case and the updated parameters can not be sent to inverters.With proposed strategy (Figure 10a, 10b), even when the loads and line parameters change to different values, the real and reactive power sharing are still good with line impedance estimation.In figure 10e and 10f, the estimation block can instantly give new values when line parameters change with high accuracy.This result has demonstrated its usefulness.In this case, the rated power ratio of the inverters is 1:2 and the line parameters of the two inverters are provided as in Table 4.The obtained results from the simulation are given in Figure 11.For this case, the ratio of active and reactive power is 1:1:1 for the three inverters with the load fixed at a pre-determined value.
the power-sharing due to the different voltage drop.When impedances of the lines connecting inverters to the common connection point are different, the current imbalance will appear as the load sharing error increases [1]. The heterogeneous line impedance including resistor and capacitance is not suitable for the conventional droop control with pure resistors or pure capacitance applying for the low voltage distribution [1], [22].Moreover, with the heterogeneous line impedance, the active and reactive power will relate and interact with each other, leading to difficulty for separate control [1]. As the line impedance changes due to the temperature, the installation position is no TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 19, SOÁ K7-2016 Trang 47 longer making the system more accurate response.Although the frequency droop technique can achieve an accurate real power sharing, the voltage droop technique typically results in poor reactive power sharing due to the mismatch in the impedances of the DG unit feeders and the different ratings of the DG units [22]-[24].Consequently, the problem of the reactive power sharing in islanded microgrids has received considerable attention in the literature and many control techniques have been developed to address this issue [31]-[32].A comprehensive treatment of the concept of virtual impedance to mitigate errors in the reactive power sharing is presented in [23]-[30].The treatment has focused on the mismatch at the output impedances of the closed-loop controlled inverters that are used to interface the DG units.With a proper design of the voltage controller, the closed-loop output impedances must be negligible at the steady state around the nominal operating frequency.Therefore, the virtual impedance can result in the accurate reactive power sharing.However, the analyses in [23]-[30] did not consider the mismatch in the physical impedance of the feeders, including transformers, cables, and the interface inductors associated with each DG unit.An interesting droop control strategy has been proposed in [21].The control strategy is composed of two stages including an initial conventional droop-based control stage and a synchronized compensation stage.

Figure 1 .
Figure 1.Block diagram of an islanded microgrid.Each inverter is connected to a common bus at the PCC point through the line impedance, In addition, loads of the microgrid are also connected to the common bus.The droop controller contains two control loops where the outer loop power control divides the capacity of each inverter and the inner loop

Figure 2 .
Figure 2. (a) Equivalent schematic of the inverters connected to the load, (b) Vector diagram of voltage and current V0, 0 are the nominal voltage amplitude voltage and frequency of inverter respectively; VS,  are the measured amplitude voltage and frequency of inverter, respectively; mp and mq are the active and reactive droop coefficients calculated as follows: of impedance of the lines connecting from the inverters to the common PCC is significantly different, the load sharing accuracy is difficult to achieve and the voltage adjustment is also difficult because it depends SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 19, No.K7-2016 Trang 50

Figure 3 .
Figure 3. Detail of Small signal adaptive real power sharing droop control.

Figure 4 .
Figure 4. Detail of Small signal adaptive reactive power sharing droop control.

Figure 5 .
Figure 5. a) The equivalent three-phase circuit of the inverter connected to loads.b) The equivalent single phase circuit of the inverter.

Figure 6 .
Figure 6.Process of Kalman filter.The equations for the updated time are to predict the state:

Figure 7
Figure 7 shows the structure of the SOGI-PLL.Both the adaptive filtering technique and in-quadrature phase detection technique are used in the SOGI-PLL to generate the frequency and phase outputs.This system has a double feedback loop, i.e. the frequency/phase generator provides both the phase-angle to the Park transform and the central frequency to the SOGI-QSG (Second Order Generalized Integrator -Quadrature Signal Generation).
Figures 9a and 9b have shown the performance of proposed strategy when the line parameters change.Figure9c and 9dshow the performance of conventional droop control.

Figure 9 .Figure 10 .
Figure 9. (a) Real power with the line impedance estimate, (c) Real power without the line impedance estimate; (b) Reactive power with the line impedance estimate, (d) Reactive power without the line impedance estimate; (e) Resistance estimation; (f) Inductance estimation; (g) Load voltage.

Figure 11 .
Figure 11.(a) Real power with the line impedance estimation, (b) Reactive power with the line impedance estimation, (c) Real power without the line impedance estimate, (d) Reactive power without the line impedance estimation, (e) Resistance estimation, (f) Inductance estimation, (g) Load voltage.
The measured active power outputs for the three inverters are shown in Figure13.The obtained active power for the three inverters are P1 = 945 W, P2 = 935 W, P3 = 945 W. The real power sharing errors, in this case, are very small.

Figure 14 .
Figure 14.The active power of the three inverters for Case study 2 of hardware experiment

Table 1 .
Parameters for the controllers