Unbalanced Power Sharing for Islanded Droop-Controlled Microgrids

Abstract

Studying the control strategy of a microgrid under the load unbalanced state helps to improve the stability of the system. The magnitude of the power fluctuation, which occurs between the power supply and the load, is generated in a microgrid under the load unbalanced state is called negative sequence reactive power $Q^-$. Traditional power distribution methods such as P-f, Q-E droop control can only distribute power with positive sequence current information. However, they have no effect on $Q^-$ with negative sequence current information. In this paper, a stationary-frame control method for power sharing and voltage unbalance compensation in islanded microgrids is proposed. This method is based on the proper output impedance control of distributed generation unit (DG unit) interface converters. The control system of a DG unit mainly consists of an active-power-frequency and reactive-power-voltage droop controller, an output impedance controller, and voltage and current controllers. The proposed method allows for the sharing of imbalance current among the DG unit and it can compensate voltage unbalance at the same time. The design approach of the control system is discussed in detail. Simulation and experimental results are presented. These results demonstrate that the proposed method is effective in the compensation of voltage unbalance and the power distribution.

I. INTRODUCTION

The microgrid is emerging as a new way to integrate distributed generation units (DG units), which can provide more reliable electric service and higher power quality to customers [1]. Three-phase DC-AC inverters are usually used as the power electronic interface to connect a DG unit and a microgrid, through which the flexibility and reliability of the microgrid can be obtained. In addition, parallel inverter operation is a key issue that should be addressed. The control objective is to obtain power sharing in proportion to the rated capacity of the DG unit and to maintain good power qualities such as a good voltage waveform, amplitude and frequency.

Inverters are usually controlled as voltage sources to provide voltage support. In addition, power-frequency and reactive-power-voltage droop control is adopted to obtain proper power sharing without critical communications among DG units, which may have a long distance between each other [2]-[6]. However, the input and output of traditional droop control become fluctuant when an unbalanced impedance is introduced into the microgrid. The power that represents the fundamental negative sequence (FNS) current cannot be proper distributed between DG units under fluctuant load unbalanced state.

A virtual synchronous generator (VSG) balance control method based on negative sequence voltage suppression has been presented in [7]. In [8], [9], a modification of VSG control based on double decoupled synchronous reference frame (DDSRF) decomposition, SG negative-sequence voltage compensation and modified calculation methods for the power and voltage is proposed. They all compensated for the negative sequence voltage. However, their power distribution is not effective.

In [10]-[12], repetitive control is introduced to compensate for voltage unbalance. However, repetitive control always has a delay of about one cycle, which reduces the response rate of the control system. On the other hand, the power distribution problem under the load unbalanced state is not addressed in these articles.

The control method presented in [13] and [14] is based on the feedback of FNS voltage. FNS reactive power Q^- is defined as the product of the FNS voltage and the FNS current. However, the definition of Q^- cannot exactly present the fluctuation of power because the FNS voltage is dramatically suppressed in well-controlled microgrid systems. In addition, the distribution of Q^- is not accurate for using Q^- as the negative feedback factor.

There is a good attempt at improving this control method in [15]. In [15], compensation of voltage unbalance at the PCC is considered. This compensation needs some specific communication lines to send messages from the PCC to each of the DG unit controllers. Thus, the control method makes the system structure more complicated and violates the “Plug & Play” rules [16].

A different definition of FNS reactive power Q^-, which refers to the product of fundamental positive sequence (FPS) voltage and FNS current, is introduced in [17]. Since the effect of the FPS voltage is more significant, this definition is more reasonable. One method for Q^- sharing through the droop control between Q^- and the output admittance has also been proposed. In this method, Q^- is accurately distributed only when the cutoff frequency is in a very high level. However, this is difficult to realize in hardware. Thus, the compensation of unbalance voltage and the accurate distribution of Q^- cannot be achieved at the same time using the control method proposed in [17] and [19].

In this paper, a model of a voltage source inverter is established in the α-β stationary reference frame and the voltage and current controllers are designed. Traditional droop control is illustrated and its inherent problems are pointed out with unbalanced loads in a microgrid. A PR controller is adopted to suppress voltage unbalance. A proportional relationship (Q^--Z control method) between Q^- and the negative output impedance Z is established to distribute the negative reactive power Q^- accurately and to reduce the effect of unbalanced loads at the same time. Finally, the proposed control strategy is verified by the simulation and experimental results.

This paper is arranged as follows. Sections II describes the model and control of the voltage source inverters. Section III illustrates the traditional droop control and its inherent problems with unbalanced loads in microgrids. Section IV describes the details of the Q^--Z control method and the whole control structure. Section V and VI show simulation and experimental results to validate the proposed control strategy. Finally, section VII presents some conclusions.

II. INNER CONTROL DESIGN

The whole microgrid structure analyzed in this paper is shown in Fig. 1. It contains two parallel-connected voltage source inverters which represent the DG unit and two static switches to decide whether the inverters are connected to the microgrid. The local and remote loads represent the power demand in the microgrid. The microgrid is operated in the stand-alone mode.

Fig. 1. Microgrid structure.

The voltage source inverter (VSI) chosen as the power electronic interface is shown in Fig. 2. It consists of a three-phase full-bridge inverter and an LC filter. The DC-AC inverter is always controlled as a voltage source, and its control objective is to track the reference voltage value and to suppress disturbances of the load current.

Fig. 2. Voltage source inverter.

The local and remote loads in the microgrid have the same structures which consist of three phase balanced loads and a 1 phase loads as shown in Fig. 3. A 22Ω resistance is connected between phase A and phase C to imitate the load unbalance state when the switch is closed.

Fig. 3. Three phase unbalanced load circuit.

The model and control of the voltage source inverter is in the α-β stationary reference frame, where no coupling exists between the α axis and the β axis. A control block diagram is shown in Fig. 4.

Fig. 4. Control block diagram of a VSI.

The current controller is:

\(G_i=k\)       (1)

where k is the current proportional coefficient. The only goal of the current controller is to increase the system damping.

It is well known that the frequencies of the fundamental frequency of the positive and negative voltages are both 50Hz in the α-β stationary reference frame. For accurate tracking of the voltage reference, a quasi resonant controller is chosen which can provide a very high amplitude at a specified frequency for an open loop Bode diagram. The voltage controller is:

\(G_{v}=k_{p v}+\frac{2 k_{v} \omega_{c} s}{s^{2}+2 \omega_{c} s+\omega_{0}^{2}}\)       (2)

where K_pv is the voltage proportional coefficient, K_v is the resonant coefficient, ω_c is the cutoff frequency, and ω₀ is the resonant frequency.

The output voltage in Fig. 4 is:

\(V=G(s) V_{\text {ref }}-Z_{\mathrm{o}}(s) i_{\mathrm{o}}\)       (3)

where V_ref is the reference voltage, G(s) is the closed-loop transfer function, i₀ is the load current, and Z₀(s) is the output impedance. In addition, G(s) and Z_o(s) are:

\(G(s)=\frac{G_{\mathrm{v}} G_{\mathrm{i}} G_{\mathrm{PWM}}}{L C s^{2}+\left(G_{\mathrm{v}}+C s\right) G_{\mathrm{i}} G_{\mathrm{PWM}}+R C s+1}\)       (4)

\(Z_{\mathrm{o}}(s)=\frac{L s+R+G_{\mathrm{i}} G_{\mathrm{PWM}}}{L C s^{2}+\left(G_{\mathrm{v}}+C s\right) G_{\mathrm{i}} G_{\mathrm{PWM}}+R C s+1}\)       (5)

In practice, the design procedure of the PR parameters can be divided into three steps. First, the bandwidth of the controller is determined by the frequency fluctuation range of the power grid so that ω_c is selected. Next, the frequency characteristics of the open-loop transfer function of the controller are selected at the gain of the fundamental frequency . In order to achieve the stability and anti-interference of the system, the K_pv parameter is designed based on the harmonic impedance.

The parameters of the voltage and current controllers are then chosen as K_pv=0.025, K_v=25, ω_c=4rad/sec, ω₀=314rad/sec and k=0.1. Then a Bode diagram of the closed loop of the voltage is shown in Fig. 5.

Fig. 5. Bode diagram of the closed voltage loop.

Then a Bode diagram of the output impedance Z₀(s) is shown in Fig. 6.

Fig. 6. Bode diagram of the output impedance Z₀(s).

From Fig. 6 it can be seen that the amplitude of the output impedance at 50Hz is about -30dB. This can effectively suppress the FPS and FNS current disturbances at the fundamental frequency.

III. TRADITIONAL DROOP CONTROL

The principle of the traditional droop control is to emulate an inverter as a synchronous generator, which decreases the frequency when the output active power increases. The equation of droop control is:

\(f=f_{0}-k_{P}\left(P-P_{0}\right)\)       (6)

\(E=E_{0}-k_{\mathrm{Q}}\left(Q-Q_{0}\right)\)       (7)

where f₀ and E₀ are the nominal frequency and voltage. P₀ and Q₀ are the nominal active and reactive power. P and Q are the instantaneous active and reactive power. f and E are the frequency and voltage reference. Then the droop characteristic is shown in Fig. 7.

Fig. 7. Droop control characteristic of active and reactive power.

Using droop control, the active and reactive power can be shared without critical communications between the inverters. According to the instantaneous reactive power theory, P and Q are calculated as follows in practical applications:

\(\left[\begin{array}{l} p \\ q \end{array}\right]=\left[\begin{array}{cc} v_{d} & v_{q} \\ v_{q} & -v_{d} \end{array}\right]\left[\begin{array}{l} i_{d} \\ i_{q} \end{array}\right]=\left[\begin{array}{c} v_{d} i_{d}+v_{q} i_{q} \\ v_{q} i_{d}-v_{d} i_{q} \end{array}\right]\)       (8)

p and q are constants when there are only linear and balanced loads in a microgrid. However, after loads becoming unbalance, the current contents FPS component and component according to the symmetrical component theory. Thus, in the rotating reference frame the output current becomes:

\(\begin{array}{l} i_{d}=I_{d}^{+}+I_{d}^{-} \cos 2 \omega t+I_{q}^{-} \sin 2 \omega t \\ i_{q}=I_{q}^{+}-I_{q}^{-} \sin 2 \omega t+I_{q}^{-} \cos 2 \omega t \end{array}\)       (9)

Substituting the current in (8) with (9), the instantaneous active p and reactive powers q become fluctuant as follows:

\(\begin{array}{l} p=V_{d}^{+} I_{d}^{+}+V_{q}^{+} I_{q}^{+}+\left(V_{d}^{+} I_{d}^{-}+V_{q}^{+} I_{q}^{-}\right) \cos 2 \omega t+\left(V_{d}^{+} I_{q}^{-}-V_{q}^{+} I_{d}^{-}\right) \sin 2 \omega t \\ q=V_{q}^{+} I_{d}^{+}-V_{d}^{+} I_{q}^{+}+\left(V_{q}^{+} I_{d}^{-}+V_{d}^{+} I_{q}^{-}\right) \cos 2 \omega t+\left(V_{q}^{+} I_{d}^{-}-V_{d}^{+} I_{d}^{-}\right) \sin 2 \omega t \end{array}\)       (10)

Note that the instantaneous active power consists of two parts. The first part is in the DC term, which contains information of the FPS voltage and current. It is as same as the active power of an inverter with balanced loads. The other part is an AC variable at a frequency of 100Hz. The amplitude of the AC variable is calculated as follows:

\(\begin{array}{l} Q^{-}=\sqrt{\left(V_{d}^{+} I_{d}^{-}+V_{q}^{+} I_{q}^{-}\right)^{2}+\left(V_{d}^{+} I_{q}^{-}-V_{q}^{+} I_{d}^{-}\right)^{2}} \\ =\sqrt{\left(V_{d}^{+2}+V_{q}^{+2}\right)\left(I_{d}^{-2}+I_{q}^{-2}\right)}=3 V^{+} I^{-} \end{array}\)       (11)

In [17], the amplitude of the AC variable is called negative reactive power Q^-, because it represents a fluctuation of the active power just like the definition of traditional reactive power. The difference is Q^-, which means the amplitude of the power exchange between the source and the loads. In addition, the traditional reactive power means the amplitude of the power exchange between loads in different phases.

Obviously, the traditional droop control can accurately distribute the DC part in (10). However, it cannot deal with the AC part—negative reactive power Q^-. Thus, the power flow cannot be shared between different DG units in a microgrid with unbalanced loads by only using traditional droop control. A Q^- distribution method is needed to solve this problem.

One point should be mentioned is that the FNS reactive power Q^- can be calculated with 3-phase traditional active power. The only requirement is a power analyzer that has 3 voltage probes, 3 current probes and is connected as shown in Fig. 8.

Fig. 8. Probe connection diagram for Q^- calculation.

Traditional 3-phase active power is calculated as follows, where U_p is the voltage peak value, \(I_p^+\) is the peak value of the FPS current, and \(I_p^-\) is the peak value of the FNS current.

\(\begin{array}{l} P_{1}=\frac{1}{2 \pi} \int_{0}^{2 \pi}\left\{U_{p} \cos \theta_{1}\left[I_{p}^{+} \cos \left(\theta_{1}+\Phi^{+}\right)+I_{p}^{-} \cos \left(\theta_{1}+\Phi^{-}\right)\right]\right\} d \theta_{1} \\ P_{2}=\frac{1}{2 \pi} \int_{0}^{2 \pi}\left\{U_{p} \cos \theta_{2}\left[I_{p}^{+} \cos \left(\theta_{2}+\Phi^{+}\right)+I_{p}^{-} \cos \left(\theta_{2}+\Phi^{-}+\frac{2 \pi}{3}\right)\right]\right\} d \theta_{2} \\ P_{3}=\frac{1}{2 \pi} \int_{0}^{2 \pi}\left\{U_{p} \cos \theta_{3}\left[I_{p}^{+} \cos \left(\theta_{3}+\Phi^{+}\right)+I_{p}^{-} \cos \left(\theta_{3}+\Phi^{-}+\frac{2 \pi}{3}\right)\right]\right\} d \theta_{3} \end{array}\)       (12)

Calculating the integral term yields:

\(\begin{array}{l} P_{1}=\frac{1}{2 \pi} \int_{0}^{2 \pi}\left\{U_{p} \cos \theta_{1}\left[I_{p}^{+} \cos \left(\theta_{1}+\Phi^{+}\right)+I_{p}^{-} \cos \left(\theta_{1}+\Phi^{-}\right)\right]\right\} d \theta_{1} \\ P_{2}=\frac{1}{2 \pi} \int_{0}^{2 \pi}\left\{U_{p} \cos \theta_{2}\left[I_{p}^{+} \cos \left(\theta_{2}+\Phi^{+}\right)+I_{p}^{-} \cos \left(\theta_{2}+\Phi^{-}+\frac{2 \pi}{3}\right)\right]\right\} d \theta_{2} \\ P_{3}=\frac{1}{2 \pi} \int_{0}^{2 \pi}\left\{U_{p} \cos \theta_{3}\left[I_{p}^{+} \cos \left(\theta_{3}+\Phi^{+}\right)+I_{p}^{-} \cos \left(\theta_{3}+\Phi^{-}+\frac{2 \pi}{3}\right)\right]\right\} d \theta_{3} \end{array}\)       (13)

In practice, P₁, P₂ and P₃ can be measured with a power analyzer and all of them should be multiplied by \(2/\sqrt{3}\) to get the accurate sum of all three phases. Thus, (13) can be seen as a ternary equation. Treating \(U_p I_p^-\) as an unknown and solving the ternary equation yields:

\(Q^{-}=\sqrt{3} U_{p} I_{p}^{-}=\frac{4}{\sqrt{3}} \sqrt{P_{1}^{2}+P_{2}^{2}+P_{3}^{2}-P_{1} P_{2}-P_{2} P_{3}-P_{1} P_{3}}\)       (14)

(14) proves that Q^- means the amplitude of the power exchange between the source and the loads from the angle of the traditional power definition. In addition, (14) can be used in practice to calculate Q^- with a power analyzer.

IV. Q-Z CONTROL

A. Q^--Z Control

In a well-controlled microgrid, the FPS voltage v⁺ can usually be seen as a constant. Thus, sharing an accurate Q^- means an accurate distribution of the FNS output current I^-. Considering a microgrid that consists of two DG units and one 3 phase unbalanced load, the DG unit keeps the output voltage very well at a fundamental frequency sinewave without unbalance or harmonic. Then the FNS current is brought in when the fundamental frequency voltage acts on unbalanced loads. When only the FNS current is considered, the unbalanced loads can be substituted with FNS current sources as below.

In Fig. 9, Z₁, Z₂ refers to the output impedance of a DG unit. Obviously, the FNS current can be shared accurately if Z₁, Z₂ can be controlled in proportion. Thus, a proportional relationship is built to operate output impedance of each DG unit.

\(Z=Z_{0}+\mu_{Z}\left(Q^{-}-Q_{0}^{-}\right)\)       (15)

Fig. 9. Equivalent circuit of the FNS current sharing of a DG unit.

As shown in Fig. 10, if outputs Q^- of one of the DG units increases, the Q^--Z control orders the DG unit to increase its output impedance to decrease the FNS output current I^-. This in turn, achieves the goal of compensating the increased Q^-.

Fig. 10. Q^--Z control characteristic.

The Q^--Z control can also be shown to be effective in math. In (11), substitute I^- with (15):

\(Q^{-}=3 V^{+} I^{-}=\frac{3 V^{+} V^{-}}{Z}=\frac{3 V^{+} V^{-}}{Z_{0}+\mu_{Z}\left(Q^{-}-Q_{0}^{-}\right)}\)       (16)

(16) can be simplified as follows:

\(\mu_{Z}\left(Q^{-}\right)^{2}+\left(Z_{0}-\mu_{Z} Q_{0}^{-}\right) Q^{-}-3 V^{+} V^{-}=0\)       (17)

Solving this equation yields:

\(Q^{-}=\frac{\mu_{Z} Q_{0}^{-}-Z_{0}+\sqrt{\left(\mu_{Z} Q_{0}^{-}-Z_{0}\right)^{2}+12 \mu_{Z} V^{+} V^{-}}}{2 \mu_{Z}}\)       (18)

If the following conditions are met:

\(\left(\mu_{Z} Q_{0}^{-}-Z_{0}\right)^{2} \gg 12 \mu_{Z} V^{+} V^{-}\)       (19)

(18) can be simplified as follows:

\(Q^{-}=\frac{\mu_{Z} Q_{0}^{-}-Z_{0}}{\mu_{Z}}\)       (20)

Thus, assume that the Q^--Z control coefficients of two DG units are assigned according to the following relationship:

\(\begin{array}{l} \frac{Q_{01}^{-}}{Q_{02}^{-}}=\frac{\mu_{Z 2}}{\mu_{Z 1}} \\ Z_{01}=Z_{02} \end{array}\)       (21)

Substitute (21) into (20). Then:

\(\frac{Q_{1}^{-}}{Q_{2}^{-}}=\frac{\mu_{Z 1} Q_{01}^{-}-Z_{01}}{\mu_{Z 2} Q_{02}^{-}-Z_{02}} \cdot \frac{\mu_{Z 2}}{\mu_{Z 1}}=\frac{\mu_{Z 2}}{\mu_{Z 1}}=\frac{Q_{01}^{-}}{Q_{02}^{-}}\)       (22)

When the relationship of formula (19) is satisfied, the simplified formula (20) can be obtained. The formula (21) is a hypothetical relationship. Through a combination of formula (21) and formula (20), formula (22) can be obtained. Then the two inverters automatically assign their respective negative-sequence reactive power according to the set ratio.

Obviously, two DG units can share Q^- in proportion to their rated capacity.

The same principle can be applied for a microgrid with multiple installations of DG units. By assigning the droop coefficients of various DG units as (23), each of the DG units can share Q^- in proportion to their rated capacity.

\(\begin{array}{l} Q_{01}^{-} \mu_{Z 1}=Q_{02}^{-} \mu_{Z 2}=\cdots=Q_{0 N}^{-} \mu_{Z N} \\ Z_{01}=Z_{02}=\cdots=Z_{0 N} \end{array}\)       (23)

Each DG unit can share Q^- in proportion to their rated capacity.

B. DG Unit Control System with Q^--Z Control

The DG unit control system with Q^--Z control is shown in Fig. 11.

Fig. 11. DG unit control diagram with Q^--Z control.

First, sequence separation is used to separate the output voltage and current into FPS and FNS. P and Q are calculated with the FPS voltage and current. Then, they are transmitted into a traditional droop controller which gives the FPS voltage reference to accurately share the active and reactive power. Q^- is calculated using the FPS voltage and FNS current. The Q^--Z controller gives the FNS output impedance to accurately distribute the FNS reactive power Q^-. Then, the FNS voltage reference is calculated as the product of Z and the FNS output current. Finally, V_ref which means the sum of the FPS voltage reference and the FNS voltage reference is used as the reference of the voltage control loop and the PR controller ensure the quality of the DG unit’s output voltage.

The increase of the negative sequence impedance of the system leads to an increase of the voltage unbalance factor (VUF). In order to keep the VUF from exceeding 3%, this paper limits the output impedance Z by 0-3Ω.

V. SIMULATION RESULTS

A. Simulation with Two DG Units

In order to verify the proposed control strategy, a simulation is implemented in PSIM, and the simulation circuit is shown in Fig. 12. Two inverters are connected in parallel and the switching frequency is 10 kHz. The rated active and reactive powers of the inverters are both 4kW and 3kVar, the nominal power is P₀=2kW, Q₀=0kVar, and the droop coefficients are k_p=0.1Hz/kW and k_Q=4.43V/kVar. The only difference is the line impedance between the two inverters, one is 3mH and the other is 2mH.

Fig. 12. PSIM simulation circuit.

Results of the traditional droop control with unbalanced loads are shown in Fig. 13. It can be seen that the active and reactive powers are both fluctuant for the FNS current. In addition, the amplitude of the power fluctuation is equal to the FNS reactive power Q^- as derived previously.

Fig. 13. Instantaneous power in a microgrid with unbalanced loads.

Using the Q^--Z control to deal with the FNS reactive power Q^-, the Q^--Z control coefficients are set as follows:

\(Z=1+\left(Q^{-}-800\right) \times 2.5 \times 10^{-3}\)       (24)

Results of the Q^--Z control with different line impedances (3mH and 2mH) are shown in Fig.14. Neglecting the difference of the line impedance, the FNS reactive power of the two inverters are distributed at a ratio of 1:1.05. In addition, their active and reactive power are also well shared by using the traditional droop control.

Fig. 14. Power distribution results using the Q^- -Z control.

The voltage unbalance factor (VUF) and an output voltage waveform of inverter 2 are shown in Fig. 14. The VUF is calculated as follows:

\(V U F=\frac{v_{r m s}^{-}}{v_{r m s}^{+}} \times 100 \%\)       (25)

It can be seen that in Fig. 15, the VUFs of the two inverters are both less than 2%, which indicates that the DG unit’ output voltage unbalance is well compensated with the Q^--Z control. The VUF is suggested to be less than 3% according to IEEE1547 standard [18].

Fig. 15. VUF and voltage waveform using the Q^- -Z control.

The target ratio of the two inverter sharing is set to 1.5:1 and the line impedances are both 3mH. The Q^--Z control coefficients are set as:

\(\begin{array}{l} Z_{1}=1+\left(Q^{-}-900\right) \times 2 \times 10^{-3} \\ Z_{2}=1+\left(Q^{-}-600\right) \times 3 \times 10^{-3} \end{array}\)       (26)

The power sharing result is shown in Fig. 16.

Fig. 16. Power sharing result when the rated capacity proportion is 1.5:1.

Obviously, the Q^- distribution result is 1.45:1, which is an accurate tracking of the target ratio. Fig. 16 also shows that the Q^--Z control does not disturb the traditional droop control which is effective for the P and Q distribution.

B. Simulation with Three DG Units

In order to achieve generality with the proposed method, the simulation study of the system is implemented with 3 DG units. The simulation is implemented in PSIM, and the simulation circuit is shown in Fig. 17. The line impedances between the three inverters are 4mH, 3mH and 2mH respectively.

Fig. 17. PSIM simulation circuit.

As shown in Fig. 18, the ratio of the FNS reactive power of the three inverters is 1:1:1.05. This shows that the proposed method can reduce the influence of the line impedance on the reactive power distribution of the line and that it is compatible with the traditional droop control. In addition, the active and reactive power are well shared by using the traditional droop control.

Fig. 18. Power distribution results using the Q^--Z control.

This also shows that in Fig. 19, the VUF of the three inverters are both less than 2%. According to IEEE1547 standard [18], the VUF should be less than 3%. These results show that the DG unit’ output voltage unbalance is well compensated with the Q^--Z control.

Fig. 19. VUF and voltage a waveform using the Q^--Z control.

VI. EXPERIMENTAL RESULTS

The experimental setup is shown in Fig. 20. It contains two parallel inverters, a static switch, line impedances and unbalanced loads. A digital processor TMS320F28335 is used to deal with the control algorithms.

Fig. 20. Experimental setup.

The results of the Q^--Z control with the same DG unit’ capacity and different line impedances (2mH and 3mH) are shown in Fig. 21 and Fig. 22. \(Q_1^{-}\) and \(Q_2^{-}\), which can be calculated using (14), are 855.1Var and 787.7Var, respectively. \(Q_1^{-}\):\(Q_2^{-}\) = 1.08:1, it is close to the goal of same capacity. This also shows that the traditional power P and Q is shared well at the same time.

Fig. 21. Power of Inverter1 with different line impedances.

Fig. 22. Power of Inverter2 with different line impedances.

The voltage unbalance factor can be seen in Fig. 23. Only the voltage situation of invereter1 is shown since its VUF is larger than the VUF of inverter2. The VUF of inverter1 represents the maximum VUF in the microgrid. Fig. 20 indicates that the Q^--Z control decreases the VUF at 2.42%, which is less than the IEEE1546 suggestion of 3%.

Fig. 23. Maximum VUF with different line impedances.

The results of the Q^--Z control with a target sharing ratio of 1.5:1 and the same 3mH line impedances are shown in Fig. 24 and Fig. 25. Using (14), it can be easily calculated that \(Q_1^{-}\)=890.4Var and \(Q_2^{-}\)=625.0Var. Therefore, \(Q_1^{-}\):\(Q_2^{-}\) =1.42:1 which meets the target ratio of 1.5:1 very well. On the other hand, the proportional ratio of P and Q of the two inverters also is close to the distribution target.

Fig. 24. Power of inverter1 for a 1.5:1 distribution target ratio.

Fig. 25. Power of inverter2 for a 1.5:1 distribution target ratio.

Fig. 26 shows the voltage result of a 1.5:1 distribution target ratio when the Q^--Z control is used. The maximum VUF is 2.79%, which is less than the IEEE1546 suggestion of 3%.

Fig. 26. Maximum VUF for a 1.5:1 distribution target ratio.

Table I summarizes the experimental results of a microgrid using the Q^--Z control. Obviously, the Q^--Z control can share Q^- accurately according to the target ratio and suppress the VUF in the microgrid at the same time. One point that should be mentioned is that the increase of the maximum VUF in the microgrid is caused by the different values of Q^- for the two DG units. According to the different Q^--Z control, a larger Q^- lead to a larger Z. In addition, a larger output impedance means that the output voltage is more easily influenced by an unbalanced output current. Thus, the maximum VUF of the target ratio of 1.5:1 is larger.

TABLE I EXPERIMENT RESULT SUMMARY OF THE Q^- -Z CONTROL

VII. CONCLUSIONS

In this paper, a model of a voltage source inverter is established in the α-β stationary reference frame and voltage and current controllers are designed. The traditional droop control is illustrated and its inherent problems are pointed out when unbalanced loads are introduced into a microgrid. Since the traditional droop control cannot share Q^-, the Q^--Z droop control is introduced as a supplement to the old power sharing method. By using the Q^--Z droop control to distribute the negative sequence reactive power Q^-, the unbalanced current in a microgrid can be reasonably configured. So that it will not affect the stability of a system, the proposed method can maintain the VUF at a low level while effectively distributing Q^-. Thus, together with the traditional droop control algorithm, a complete power distribution system is proposed.