Parallel Operation of Microgrid Inverters Based on Adaptive Sliding-Mode and Wireless Load-Sharing Controls

Abstract

This study proposes a new solution for the parallel operation of microgrid inverters in terms of circuit topology and control structure. A combined three-phase four-wire inverter composed of three single-phase full-bridge circuits is adopted. Moreover, the control structure is based on adaptive three-order sliding-mode control and wireless load-sharing control. The significant contributions are as follows. 1) Adaptive sliding-mode control performance in inner voltage loop can effectively reject both voltage and load disturbances. 2) Virtual resistive-output-impedance loop is applied in intermediate loop to achieve excellent power-sharing accuracy, and load power can be shared proportionally to the power rating of the inverter when loads are unbalanced or nonlinear. 3) Transient droop terms are added to the conventional power outer loop to improve dynamic response and disturbance rejection performance. Finally, theoretical analysis and test results are presented to validate the effectiveness of the proposed control scheme.

I. INTRODUCTION

Microgrid is a low-voltage network with different microsources and distributed loads operating to supply electric power for a local area. Most microsources are interfaced through power electronic converters to provide loads with reliable and high-quality power [1]-[4]. In such systems, every power electric converter should be able to control independently without communication links because of the long distance among microsources [5], [6]. Concurrently, power electric converters should be able to share the variable distributed loads in proportion with the power ratings of converters. To realize the function of power sharing without intercommunication, droop methods that emulate the behavior of large power generators are usually adopted [7]-[11]. The basic prerequisite for applying droop methods is that the equivalent output impedance of the converter should be resistive or inductive. However, in low-voltage microgrids, output impedance is usually resistive-inductive and measuring or estimating is difficult, which makes proportional power sharing impossible [12], [13].

Output impedance is determined through circuit topology, control structure, and line impedance. A possible solution to the impedance problem is adding an inductor in a series with a converter output. However, this inductor is heavy and bulky, and causes imbalance among the three phases. Hence, another method that puts a virtual output impedance loop into the control structure is usually adopted [14], [15]. Given that line impedance is predominantly resistive in low-voltage microgrids, virtual resistive-output-impedance loop is used in this study. Transient droop terms are also added into the conventional droop control method to improve the dynamic response and disturbance rejection performance.

Problems on imbalance and harmonics are also important in the parallel operation of microgrid inverters [16], [17]. The output performance and robustness of microgrid inverters are mainly affected by the effectiveness of the control strategy [18]-[25]. In the past decade, various closed-loop control techniques were reported to achieve the dynamic characteristic and disturbance rejection performance under different types of loads, such as proportional-integral control [18], proportional-resonant control [19], [20], Lyapunov-function-based control [21], H∞ control [22], and fuzzy control [23]. However, most of these works are only suitable for three-phase balanced circuit or single-phase circuit, which could also not meet all load conditions, such as unbalanced loads and nonlinear loads. Recently, some works focused on the sliding-mode control method. In [24], a robust sliding-mode controller is proposed to control the active and reactive powers of a doubly fed induction generator wind system without involving any synchronous coordinate transformation. However, the unbalanced condition is not considered in the study. In [25], a combined fuzzy adaptive sliding-mode voltage controller is used for three-phase uninterruptible power supply (UPS) inverter. Moreover, in [26], Mohamed et al. present a direct-voltage control strategy for microgrid converters based on sliding-mode dynamic controller. All these methods can realize robust operation in isolated or grid-connected modes, but output performance is quite poor because of chattering. Hence, this study presents a voltage regulation strategy based on adaptive three-order sliding-mode control. Meanwhile, to solve the unbalanced problem, a combined three-phase four-wire inverter composed of three single-phase full-bridge circuits is adopted.

The remainder of this paper is organized as follows. Section II describes the system which involves the circuit topology and the basic principle of decentralized parallel operation. Section III presents the adaptive three-order sliding-mode voltage control with a virtual resistive-output-impedance loop and wireless load sharing control. Section IV shows the test results, which demonstrate the effectiveness and applicability of the proposed control strategies. Finally, Section V presents the conclusion.

 

II. SYSTEM DESCRIPTIONS

Fig. 1 shows a general microgrid system that consists of distributed generation (DG) units, distributed loads, and voltage source inverters (VSIs) that transfer the energy of DG units into an AC bus. VSIs generally operate in grid-connected mode, and power is transmitted from DG units into the grid. When a fault occurs or the power quality worsens in the utility, the microgrid system disconnects from the grid by cutting off switch S1 and entering intentional islanding mode. DG units and VSIs should be able to share variable distributed loads in proportion with the power ratings of the units and maintain AC bus voltage.

Fig. 1.General microgrid system.

A. Topology and Modeling

Distributed loads are usually unbalanced in actual microgrids, and sometimes single-phase loads are dominant. Hence, in terms of inverter design, the serious load imbalance problem should be considered primarily. The VSI shown in Fig. 2(a) is then adopted for the microgrid power converter in this study, which is composed of three single-phase full-bridge circuits (T1–T12), low-pass filters (Lf and Cf), and an isolated transformer (T). Rf is the per-phase resistance of the LC filter, and Zline is the impedance of the line. u and i are the output voltage and current of the modular inverter circuit respectively, while vo and io are the output voltage and current of the low-pass filter respectively. Subscripts a, b, and c represent the three phases. Each phase in the topology can also be controlled independently.

Fig. 2.Topology of the inverter and the equivalent dynamic model. (a) Topology of the three-phase four-wire inverter. (b) Equivalent dynamic model of the single phase.

Before analyzing the VSI model, the following assumptions are made: 1) the isolated transformer T is ideal, and the turn ratio of the transformer is 1:1; 2) all switching devices are ideal, and the delay time can be disregarded. Therefore, the dynamic equation of every phase in the VSI can be represented as follows:

where KPWM the equivalent parameter of the modular inverter circuit, vcon is the input control signal, and KPWM vcon represents the output voltage of the modular inverter circuit. Eqs. (1) and (2) can be represented as

Through Laplace transformation, the dynamic model of the single phase shown in Fig. 2(b) can be obtained.

B. Power Sharing Control

Fig. 3 shows the schematic diagram of a microgrid with two distributed generations. U∠0 is the AC bus voltage, and E1∠ϕ1 and E1∠ϕ2 are the output voltages of the two inverters. ϕi is the phase angle difference between the output and bus voltages. ri emulates the sum of the output and line resistances, while Xi is the sum of the output and line inductances. The active power and reactive power of inverter i can be represented as

Fig. 3Equivalent circuit of a microgrid with two inverters.

where |Zi| is the impedance amplitude of inverter i, and θi is the impedance angle.

Assuming that the impedance of the inverter is resistive (Zi = Ri), the active and reactive powers become

To realize the power sharing function, conventional droop characteristics are usually used in the parallel operation of microgrid inverters:

where ωi* and Ei* are the nominal angular frequency and voltage of the inverter respectively. m and n are the droop coefficients.

C. Output Impedance of the Inverter

Droop control and power-sharing accuracy rely on the impedance angle and amplitude respectively. However, given the existence of line and output impedance differences between the two inverters, the accurate value of the equivalent output impedance is difficult to measure or calculate [14]. In this situation, a method that adds virtual impedance loop into control action is proposed. With a suitably designed virtual impedance, the equivalent output impedance of the inverter can effectively experience either inductive or resistive, and power-sharing accuracy can be greatly improved. In [15], the relationship between equivalent output impedance and power rating of the inverter is derived in detail.

 

III. CONTROL DESIGN

This section aims to propose a controller that can guarantee parallel operation of microgrid inverters with robust performance and accurate power sharing. The proposed control structure of a microgrid inverter is shown in Fig. 4. The structure consists of three main control loops: 1) inner voltage regulation loop, 2) virtual output impedance loop, and 3) outer P/Q sharing control loop. Given the particularity of the inverter topology adopted in this paper, the inner voltage regulation loop and virtual output impedance loop are controlled independently for each phase, whereas the outer P/Q sharing control loop is calculated for all the phases lumped together.

Fig. 4Proposed control structure of the inverter.

A. Inner Voltage Regulation Loop

The main objective of the inner voltage regulation loop is to maximize the disturbance rejection performance and have an excellent voltage tracking performance. Based on Eq. (3), the state equation of the single phase can be represented as follows:

where x(t) = vo, u(t) = vcon, ap = −Rf/Lf, bp = −1/CfLf, cp = KPWM/CfLf, and n(t) represents the sum of all uncertainties caused by parameter variation and dynamic and load disturbances. n(t) is assumed to be bounded (|n(t)|<ρ).

Define a voltage tracking error e = vo-vcmd and a three-order dynamic sliding surface s(t) as

where vo is the system output voltage, vcmd is the reference voltage command, and k1 and k2 are nonzero positive constants.

As shown in Fig. 5, the proposed control scheme in every phase is composed of three parts: equivalent model controller, switching controller, and adaptive observation. The function of the equivalent model controller is to specify the desired performance based on the inverter model, and the output voltage of this controller is utr. The objective of switching controller is to suppress uncertainties and unpredictable perturbation to ensure the equivalent model controller performance, and the output voltage is usw. Adaptive observation is designed to alleviate the chattering phenomenon, which is inevitable in the sliding-mode control method. By estimating the upper bound of the uncertainties, the observation can choose the control gain adaptively. According to the dynamic model, the control law can be designed as follows:

Fig. 5Block diagram of the output voltage closed loop based on adaptive sliding-mode control

where λ is a positive constant.

To prove the voltage control law, a Lyapunov function candidate is defined as follows:

where The derivative of the Lyapunov candidate function is

According to Eqs. (12)–(17), the following can be obtained:

Based on the analysis above, the stable behavior of the adaptive three-order sliding-mode voltage control can be ensured, and the proposed control scheme has no strict requirement for the model parameters. The method of control parameter selection and the analysis of the inverter equivalent output impedance are then described in the following.

By comparing Eqs. (3) and (12), Eq. (19) can be obtained:

Through Laplace transformation, Eq. (19) can be represented as follows:

where Zo(s) is the output-impedance transfer function. Fig. 6 shows the equivalent circuit of the inverter. Based on Eq. (20), the dynamics of the output voltage is affected by the output impedance of the inverter, and the desired dynamic response can be obtained by adjusting the system poles with the suitable selection of k1 and k2. Fig. 7 shows the root locus for different k1 and k2 values. Evidently, the poles gradually come close to an imaginary axis as k2 decreases, hastening the system but making it more oscillatory. In comparison, when k1 is increased, the poles move farther away from the real axis, resulting in a less damped system.

Fig. 6.Equivalent circuit of the inverter with the inner voltage regulation loop.

Fig. 7.Root-locus diagrams. (a) k1=9×109 for 0≤ k2≤2×105. (b) k2=1.4×105 for 5×109≤k1≤1×1010.

Table I lists the detailed parameters of the 3φ four-wire inverter. The bode diagram of the output impedance can then be obtained, as shown in Fig. 8. The output impedance value clearly has comparable resistive and inductive terms at 50, 150, 250, 350 Hz, and so on. For example, at the power frequency (50 Hz), the output impedance is about −50 dB, and 80 deg, whereas if line impedance is considered, the output impedance is about −30 dB and 10 deg.

TABLE IDETAILS OF THE 3Φ FOUR-WIRE INVERTER

Fig. 8.Bode diagram of the output impedance with and without the line impedance.

The phase and amplitude of output impedance is very sensitive to line impedance. Nevertheless, line impedances are difficult to measure and estimate in an actual system; line impedances are usually different among various inverters and even differ among the three phases of an inverter. Thus, power-sharing accuracy among parallel inverters is not guaranteed.

B. Virtual Resistive-Output-Impedance Loop

To meet the requirements of parallel operation for microgrid inverters, virtual output impedance loop is added into the control structure. By dropping the output voltage reference v*cmd proportionally to the output current, as shown in Fig. 4, the equivalent output impedance of the closed-loop inverter can be changed and fixed. The input reference voltage of the inner loop can then be rewritten as

where Zo*(s)= Rd + Zo(s) is the new equivalent output impedance of the inverter, and v*cmd is the voltage reference at no load. Fig. 9 shows the influence of Rd on output impedance. Increasing the value of Rd leads to increasingly resistive output impedance at the frequencies of 50, 150, 250, 350 Hz, and so on. Concurrently, the magnitudes of the output impedance at such frequencies tend to 20 lg Rd. Apparently, as Rd increases, the values of the original output and line impedances can be neglected. However, excessive Rd value would reduce the voltage reference considerably and cause the steady-state error of the system to increase. Accordingly, through proper design of the Rd value, the power-sharing accuracy of the parallel operation for inverters can be ensured, regardless whether the loads are balanced, unbalanced, or nonlinear.

Fig. 9.Bode diagram of the output impedance with Rd variation (from 0 to 1).

C. Outer P/Q Sharing Control Loop

As shown in Fig. 10, the outer P/Q sharing control loop can be divided into three parts: power calculation, modified P/Q droop, and reference voltage generation. Given the existence of unbalanced loads, the method to calculate the active and reactive powers for each phase is adopted. The instantaneous active and reactive powers can be expressed as follows:

Fig. 10.Block diagram of the power-sharing controller.

where vo and io are the measure values of the output voltage and current for each phase respectively, and H(io) demonstrates the Hilbert transform of io. Next, p and q should be processed by low-pass filters (LPFs). Subsequently, the total active and reactive powers are the sum of the three phases. In the figure, subscripts a, b, and c represent the three phases.

In microgrid dynamics, low-frequency oscillation modes generated by power-sharing controllers and power filters are dominant [25]. To enhance the performance of a conventional droop controller, transient droop terms can be added into Eq. (8). Transient droop functions increase the controllability of the power-sharing controller by adding a second degree of freedom in control turning. The modified droop functions are given by

where md and nd are transient droop coefficients. The proposed control method allows transient response to be modified by acting on the main control parameters while maintaining the static droop characteristics. In addition, the proposed method minimizes the transient circulating current among the modules and further improves the whole system dynamic performance. Coefficients m and n fix the steady-state control objectives, while md and nd are selected to guarantee stability and excellent transient response.

To investigate the stability and transient response of the system, small-signal analysis is performed [27]. Considering the effect of LPFs, the small-signal dynamics of active and reactive powers [Eqs. (6) and (7)] can be expressed as

where ωc/(s+ωc) is the LPF, ωc is the cutoff frequency, and denote the perturbed values of Ei and ϕi respectively. By differentiating Eqs. (24) and (25), the following are obtained:

According to Eqs. (26)–(29), the following dynamics can be obtained:

The characteristic equation of the close-loop system can then be obtained as follows:

where

A = Ri + mdωcU cosϕi

B = ωc[2Ri + (ndEi + m + mdωc)U cosϕi + mdndωcEiU2/Ri]

C = ωc[Riωc + (nEi + ndωcEi + mωc)U cosϕi + (mnd+mdn)ωcEiU2/Ri]

D = ωc2nUEi (cosϕi + mU/Ri)

According to the characteristics in Eq. (32), system poles can be fixed. System stability and desired dynamical response can then be obtained by adjusting these poles with suitable selection of md and nd. Using the inverter parameters listed in Table I, the root-locus diagrams for different md and nd values are illustrated in Fig. 11. Fig. 11(a) clearly reveals that by decreasing nd, system poles will be close to an imaginary axis, making the system become oscillatory and even unstable. Fig. 11(b) indicates that by increasing md, first-order dynamics will be increasingly dominant, whereas decreasing md makes the second-order dynamics become dominant. Hence, transient droop coefficients can be obtained.

Fig. 11.Root-locus diagrams. (a) md=1×10-5 for 0≤ nd≤1×10-5. (b) nd=1×10-6 for 0≤ md≤2×10-4.

 

IV. RESULTS

This section evaluates the performance of the proposed controller and the parallel operation for the microgrid system depicted in Fig. 1 through simulation and experiment. The test involves two DG units. Both the topologies adopt combined three-phase inverter circuits, as shown in Fig. 2. The circuit and control parameters are presented in Table 2. Fig. 12(a) shows the experimental setup. Only single-phase full-bridge circuits are used in the inverters because of the particularity of the combined three-phase four-wire topology. The design capacities are 5 and 2.5 kW. The composition of INV1 is shown in Fig. 12(b). Depending on the loads, the results of the three cases are discussed in the following sections:

Fig. 12.Experimental setup.

TABLE IIPARAMETERS OF THE INVERTERS FOR PARALLEL OPERATION

A. Case 1: Resistive Load

In this scenario, the simulation and experiment results of sudden load increase and reduction are considered. The initial load of every phase is set as 3 kW, and the load change value is 2 kW.

Figs. 13 and 14 show the simulation results of the parallel inverters. The commands of sudden load increase and reduction are set in t = 0.1 and 0.3 s respectively. In Fig. 13(a), symbols 1 and 2 represent the output currents of INV1 and INV2 respectively. Obviously, the inverters share the load current proportionately to its power ratings, and the control performance of the transient dynamics is very well that the currents can stabilize in one circle. Fig. 13(b) indicates that output voltage can remain stable when load changes, and the total harmonic distortion (THD) is less than 5%. However, given the existence of virtual resistive-output-impedance loop, very small variations (t = 0.1 and 0.3 s) occur on the output voltage and are almost negligible here. Fig. 14 shows the instantaneous power for load and parallel inverters. P and Q represent the active and reactive powers. Evidently, load power can be shared in proportion with the power rating of the inverter.

Fig. 13Simulation results under resistive loads. (a) A-phase currents of the load, INV1 and INV2. (b) Load voltage.

Fig. 14.Simulation results under resistive loads. (a) Active power (upper trace) and reactive power (lower traces) responses of the loads. (b) Power responses of INV1. (c) Power responses of INV2.v

The experiment results are shown in Fig. 15. The upper trace is load voltage, and the lower traces are output currents of the two inverters. The output current of INV1 is nearly double of INV2 regardless of whether the inverters operate in steady or transient state. Meanwhile, voltage stability is unaffected by the transient state.

Fig. 15.Experiment results under resistive loads: load voltage (upper trace) and output currents (lower traces) of INV1 and INV2.

The simulation and experiment results show that the proposed control structure is effective for parallel inverters under resistive loading condition, and power-sharing accuracy and dynamic response can be ensured.

B. Case 2: Resistive-Inductive Load

Resistive-inductive loading condition with a sudden change is considered in this scenario.

The simulation results are shown in Figs. 16 and 17. The initial loads of every phase are set as P = 3 kW, Q = 1 kVar, and the load change values are P = 1 kW, Q = 2 kVar. The variation commands of the loads are set in t = 0.1 and 0.3 s. Fig. 16 shows the output voltage and current responses of the two inverters. The output currents of INV1 and INV2 are proportional to the power rating of both inverters, and resistive-inductive load cannot affect the power-sharing accuracy and stability of the output voltage. Meanwhile, the current dynamics can stabilize within two circles. Fig. 17 shows the performance of the instantaneous power. Compared with the resistive loading condition, the transient response is longer.

Fig. 16.Simulation results under resistive-inductive loads. (a) A-phase currents of loads INV1 and INV2. (b) Load voltage.

Fig. 17.Simulation results under resistive-inductive loads. (a) Active power (upper trace) and reactive power (lower traces) responses of the loads. (b) Power responses of INV1. (c) Power responses of INV2.

To test the robustness of the proposed control structure, pure inductive loads are added to the system in the experiment, as shown by the results in Fig. 18. The transient voltage disturbances are effectively rejected because of the robust sliding-mode control performance. Furthermore, the power sharing accuracy between the two inverters is ensured.

Fig. 18.Experiment results under resistive-inductive loads: load voltage (upper trace) and output currents (lower traces) of INV1 and INV2

C. Case 3: Unbalanced and Nonlinear Loads

Unbalanced and nonlinear loading conditions are considered in this scenario.

To test the robustness of the proposed control strategies in rejecting unbalanced disturbances, unbalanced loads are added into the system in the time period 0.1 s < t < 0.3 s. The simulation performance of the load voltage and current under unbalanced loads are shown in Figs. 19(a) and 19(b). The three phase current amplitudes and phases differ, but the load voltage remains balanced. The three-phase voltages are balanced because of the special topology used in this study. The output current waveforms of INVs are shown in Figs. 19(c) and 19(d). The current amplitudes of INV1 and INV2 are proportional to the power ratings of both inverters. The current phases of INV1 and INV2 are also the same; thus, power-sharing accuracy under unbalanced loading condition is ensured.

Fig. 19.Simulation results under unbalanced loads. (a) Load voltage. (b) Load current. (c) INV1 output current; and (d) INV2 output current.

To test the robustness of the proposed control strategies under nonlinear loading condition, the uncontrolled rectifier circuit is used, and load power is set to 10 kW. The simulation results are shown in Fig. 20. Fig. 20(a) shows the load voltage waveform, and the frequency spectra of this waveform are analyzed in Fig. 20(c). The output voltage, which yields a THD of 1.21%, is regulated to reject nonlinear load disturbances. The load and output currents of the INVs are shown in Fig. 20(b). The frequency spectra are correspondingly analyzed in Figs. 20(d)–20(f). The THDs of these currents (35%, 34.95%, and 35.19%) are nearly the same.

Fig. 20.Simulation results under nonlinear loads. (a) Load voltage. (b) A-phase currents of the load, INV1 and INV2. (c) Load voltage spectra. (d) Load current spectra. (e) Output current spectra of INV1. (f) Output current spectra of INV2.

In the experiment, the load power of the single-phase uncontrolled rectifier circuit is set to 1 kW; the results are shown in Fig. 21. Compared with the simulation results, the load voltage and output currents have higher THDs. However, the power-sharing accuracy is maintained. The inverters share not only the fundamental currents, but also the harmonic currents. Thus, power-sharing accuracy under nonlinear loading condition is ensured.

Fig. 21.Experiment results under nonlinear loads: load voltage (upper trace) and output currents (lower trace) of INV1 and INV2.

The aforementioned results show that the proposed adaptive sliding-mode control and dynamic load sharing control can be used effectively and reliably under different loading conditions for the parallel operation of microgrid inverters.

 

V. CONCLUSION

A new solution for the parallel operation of microgrid inverters in terms of circuit topology and control structure is proposed in this study. The three-phase four-wire inverter composed of three single-phase full-bridge circuits is adopted. The control structure consists of three nested loops: 1) inner voltage regulation loop, 2) virtual output impedance loop, and 3) outer P/Q sharing control loop. Adaptive sliding-mode control is used in the inner voltage loop for the output voltage of the inverter to effectively reject load disturbances, regardless whether loads are balanced, unbalanced, or nonlinear. In a precise contrast with the conventional droop method and the actual low-voltage microgrid, virtual resistive-output-impedance loop is used to enforce the equivalent output impedance of the inverters to be resistive and proportional. Finally, a new wireless power-sharing control method that involves transient droop terms is adopted in the outer P/Q sharing loop. Consequently, excellent power-sharing accuracy can be achieved for all kinds of loads: balanced, unbalanced, or nonlinear. Meanwhile, system stability and dynamic response can also be improved. Finally, theoretical analysis and test results verify the effectiveness and superiority of the proposed control structure.