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A control strategy for microgrid inverters based on adaptive three-order sliding mode and optimized droop controls
Electric Power Systems Research 117 (2014) 192–201
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A control strategy for microgrid inverters based on adaptive three-order sliding mode and optimized droop controls Yancheng Liu, Qinjin Zhang ∗ , Chuan Wang, Ning Wang Marine Engineering College, Dalian Maritime University, Dalian 116026, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 9 April 2014 Received in revised form 12 August 2014 Accepted 25 August 2014 Available online 16 September 2014 Keywords: Microgrid inverter Distributed generation (DG) Adaptive three-order sliding-mode control Droop control Disturbance rejection Nonlinear load
a b s t r a c t Robust control and seamless formation are the two crucial problems that affect smart microgrids. This paper proposes a new solution for microgrid inverters in terms of circuit topology and control structure. The combined three-phase four-wire inverter, which is composed of three single-phase full-bridge circuits, is adopted in this study. The control structure is based on the inner adaptive three-order slidingmode closed-loop, the immediate virtual output-impedance loop, and the outer power control loop. Three significant contributions are obtained: (1) the microgrid inverters effectively reject both voltage and load disturbances with the adaptive sliding-mode controllers regardless of whether the inverters are operating in the grid-connected mode, islanding mode, or transition from the grid-connected mode to the islanding mode; (2) the virtual output impedance loop is applied to make a resistive equivalent output impedance of the inverters and to meet the requirements of the inverter parallel operation in the islanding mode; (3) the proposed droop method reduces the line inductive impedance effects and improves the power sharing accuracy by optimizing the droop coefficients. The theoretical analysis and test results validate the effectiveness of the proposed control scheme. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The installation of distributed generations (DGs) has been given progressive attention not only because of the increasing concern toward environmental emissions from centralized power plants but also because of economic and technical reasons [1–4]. However, the penetration of DGs in a power system is limited because of technical reasons, such as stability constraints. IEEE Standard 1547.4 has recently proposed a clustering concept that uses a microgrid as the building block of DG power systems [5–7]. The microgrid should be able to operate in both the grid connected and islanding modes. Furthermore, the transition between the two modes should be seamless. DGs are usually interfaced through power electronic converters to provide loads. A flexible operation and a robust control of DG interfaces are the major objectives of smart microgrid research [6,7]. The output performance and the robustness of microgrid inverters are mainly affected by the effectiveness of the control strategy used [8–13]. The control strategy is usually composed of the outer power control loop and the inner voltage closed-loop. In the past
∗ Corresponding author. Tel.: +86 13664290653. E-mail addresses: [email protected] (Y. Liu), [email protected] (Q. Zhang), [email protected] (C. Wang), [email protected] (N. Wang). http://dx.doi.org/10.1016/j.epsr.2014.08.021 0378-7796/© 2014 Elsevier B.V. All rights reserved.
decade, various closed-loop control techniques (e.g., proportionalresonant control [9], Lyapunov-function-based control [10], two degrees-of-freedom control [11], H∞ control [12], and fuzzy control [13]) have achieved a dynamic characteristic and disturbance rejection under different load types. However, most of these works are only suitable for the grid-connected mode or islanding mode. Some recent works have focused on the sliding-mode control method [14–17]. A robust sliding-mode controller is proposed in Ref. [14] to control the active and reactive powers of a double fed induction generator (DFIG) wind system without involving any synchronous coordinate transformation. However, the DFIG wind system usually operates in the grid-connected mode and stops when a fault occurs or the power quality in the utility decreases. Wai et al. [15] design the adaptive total sliding-mode controllers, which correspond to different operation modes for single-phase inverters. However, the parallel operation has not been considered. Mohamed et al. [16] present a direct-voltage control strategy for a microgrid converter on the basis of the sliding-mode dynamic controller. The method can realize a normal operation in both the grid-connected and isolated modes. However, the output current performance in the grid-connected mode is unimpressive because of the indirect current control. In Ref. [17], a dual-loop controller is proposed for a voltage source inverter control. The inner loop is designed by using the sliding-mode control strategy, and the inner loop generates the pulse-width modulation voltage
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parallel operation. Section 3 presents the proposed control structure and the analysis. Section 4 shows the test results, which demonstrate the effectiveness and the applicability of the proposed control strategies. Section 5 concludes. 2. System descriptions
Fig. 1. A simplified microgrid system.
commands to regulate the inverter currents. Nevertheless, previous studies have not considered the chattering phenomenon of the sliding-mode control. The current study designs new closed-loop controllers that correspond to different operation modes on the basis of the adaptive three-order sliding-mode control. The closedloop adopts direct current control in the grid-connected mode, whereas the islanding mode adopts the direct-voltage control. The voltage and load disturbances are effectively rejected. Furthermore, the dynamic switching process from the grid-connected mode to the islanding mode is smooth. In order to solve the unbalanced problem, the combined three-phase four-wire inverter, which is composed of three single-phase full-bridge circuits, is adopted in the paper. Another problem for microgrid inverters in the islanding mode is the power sharing accuracy. The inverters should be able to proportionally share the distributed loads to their power ratings. The droop methods, which emulate the behavior of large power generators, are usually adopted to wirelessly perform functions [19–21]. The equivalent output impedance of the inverter should be resistive or inductive, which is the basic prerequisite for the application of the droop method. In the actual system, the output impedance is resistive–inductive and hard to measure or estimate. A possible solution is the addition of an inductor in the series with an inverter output. Nevertheless, this inductor is heavy and bulky and can cause an imbalance among the three phases. Hence, another method that places the virtual output impedance loop into the control structure is adopted [22,23]. This paper adopts a virtual resistive output impedance loop on this basis. Furthermore, an optimized droop controller that considers line impedance is designed. The power sharing accuracy and dynamic response can be effectively improved by controlling the current feedback gain and optimizing the droop coefficients. Section 2 describes the system, including the circuit topology, the system model, and the basic principle of the decentralized
A simplified microgrid system that consists of DGs, distributed loads, utility, voltage source inverters (VSIs) and AC bus is shown in Fig. 1. VSIs generally operate in the grid-connected mode, and the power is transmitted from the DGs into the utility. When a fault occurs or when the power quality in the utility worsens, the microgrid system disconnects from the utility by cutting off switch S1. The system then enters the intentional islanding mode. DGs and VSIs should be able to proportionally share the variable distributed loads to their power ratings and maintain the consistency of load voltage. If the utility recovers, the microgrid system reconnects to the utility. The dynamic models of the inverter in the grid connected and islanding modes differ according to the operation mode performance. 2.1. Inverter topology In the actual microgrid, the distributed loads are usually unbalanced, and sometimes the single-phase loads are dominant. So when it comes to the inverter design, the serious load imbalance problem should be considered firstly [18]. Then the VSI topology which is shown in Fig. 2 is adopted in this paper. The topology is composed of three single-phase full-bridge circuits (T1–T12), lowpass filters (Lf and Cf ), and an isolated transformer (T). Rf is the LC filter resistance per phase, and Zline is the line impedance between the inverter and bus. u and i are the output voltage and the current of the modular inverter circuit, respectively. vo and io are the output voltage and current of the low-pass filter. Subscripts a, b, and c represent the three phases. Furthermore, each phase in the topology can be controlled independently. 2.2. The model in the grid-connected mode Before analyzing the model, the following assumptions are made: (1) all switching devices are ideal and delay time is neglected; (2) the isolated transformer T is ideal, its turn ratio is 1:1 and its phase angle shift is not considered. Therefore, when the microgrid operates in the grid-connected mode, the dynamic equation of every phase in the VSI is represented as follows: dio 1 = (KPWM vcon − vs − io Rf − vsd ), Lf dt
Fig. 2. Topology of the three-phase four-wire inverter.
(1)
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is the sum of the output inductance and line inductance. The active and reactive powers of the inverter i are represented as follows: 1
Pi =
[(UEi cos i − U 2 ) cos i + UEi sin i sin i ], Zi
Qi =
[(UEi cos i − U 2 ) sin i − UEi sin i cos i ], Zi
1
(6)
(7)
where Zi is the impedance amplitude of the inverter i, Zi =
ri2 + Xi2 , and i is the impedance angle. Assuming that the impedance of the inverter is resistive (Zi = Ri ), the active and reactive powers are expressed as follows:
Fig. 3. VSI model (a) in the grid-connected mode and (b) islanding mode.
Pi =
1 (UEi cos i − U 2 ), Ri
Qi = −
UEi sin i . Ri
(8)
(9)
The conventional droop characteristics are used to conduct the parallel operation function:
ωi = ωi∗ − n(Qi∗ − Qi )
Fig. 4. Schematic of a microgrid with two DGs.
where KPWM is the equivalent parameter of the modular inverter circuit, vcon is the input control signal, KPWM vcon represents the output voltage of the full-bridge circuit, vs is the grid voltage, and vsd emulates the grid voltage disturbance. Eq. (1) can be represented as follows: Rf KPWM 1 1 i˙ o = vcon − vs − io − vsd . Lf Lf Lf Lf
(2)
The dynamic model of the VSI in the grid-connected mode is shown in Fig. 3(a). Laplace transformation is used in this model. 2.3. The model in the islanding mode When the microgrid operates in the islanding mode, the dynamic equation of every phase in the VSI is represented as follows: diLf dt
=
1 (KPWM vcon − vo − iLf Rf ), Lf
dvo 1 = (i − io ), Cf Lf dt
(3)
(4)
where vo is the output voltage of the L and C filters, and iLf is the inductance current. Eqs. (3) and (4) are then represented as follows:
v¨ o =
Rf Lf
v˙ o −
Rf 1 K 1 vo + PWM vcon − i˙ o − io . Cf Lf Cf Lf Cf Cf Lf
Ei = Ei∗ + m(Pi ∗ − Pi )
(5)
The dynamic model of the VSI in the islanding mode is shown in Fig. 3(b). Laplace transformation is used in this model. 2.4. Power sharing control The schematic of a microgrid with two DGs is shown in Fig. 4. U ∠ 0 is the AC bus voltage, E1 ∠ 1 and E2 ∠ 2 are the output voltages of the two inverters, is the phase angle difference between the output voltage and bus voltage. ri emulates the sum of the output resistance and line resistance of the inverter i (i = 1 or 2), and Xi
,
(10)
where ωi∗ and Ei∗ are nominal angular frequency and inverter voltage, respectively. m and n are droop coefficients. In the actual system, the equivalent output impedance of the inverter is hard to measure or calculate. In the low-voltage microgrid, the line impedance is usually resistive–inductive and has a value that is proportional to the length of the line. Under this circumstance, the resistance or the induction should not be neglected. Hence, the conventional droop method has a few limitations.
3. Proposed control scheme A new control structure for the three-phase four-wire inverter, which can operate in both the grid connected and the islanding modes, is proposed. The proposed control structure is shown in Fig. 5. The structure consists of six main parts: (1) the sliding-mode current inner loop; (2) sliding-mode voltage inner loop; (3) virtual output impedance medium loop; (4) constant power control outer loop; (5) P/Q sharing control outer loop; (6) islanding detection and mode selection. The microgrid inverter automatically operates in the grid connected or islanding mode by using the islanding detection and mode selection unit. In the grid-connected mode, the inverter is controlled by the sliding-mode current inner loop and constant power control outer loop. The inverter effectively rejects the grid voltage disturbances and parameter uncertainties. Moreover, the inverter provides a constant current to the grid. In the islanding mode, the inverter is controlled by the sliding-mode voltage inner loop, virtual output impedance medium loop, and P/Q sharing control outer loop. The controller guarantees the parallel operation of the microgrid inverters with robust performance and good power sharing accuracy. vs is the measure value of the grid voltage, and vo and io are the output voltage and the inverter current, respectively (Fig. 5). P* and Q* are the given values of the active and the reactive powers, respectively. The inner closed-loop and virtual output impedance medium loop are independently controlled for each phase, whereas the outer power loop is calculated for all lumped phases because of the particularity of the inverter topology adopted in this paper.
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Fig. 5. Proposed inverter control structure.
3.1. The inner closed-loop in the grid-connected mode
3.2. Inner closed-loop in the islanding mode
The main objective of the inner closed-loop in the gridconnected mode is to maximize the voltage disturbance rejection performance and obtain a good current tracking performance. From Eq. (2), the state equation of every phase is represented as follows:
The main objective of the inner closed-loop in islanding mode is to maximize the load disturbance rejection performance and obtain a good voltage tracking performance. From Eq. (5), the state equation of every phase is represented as follows:
x˙ g (t) = ap xg (t) + bp u(t) + cp z(t) + m(t)
˙ + fp x(t) + gp u(t) + n(t), x¨ (t) = dp x(t)
(11)
where xg (t) = io , u(t) = vcon , z(t) = vs , ap = − Rf /Lf , bp = KPWM /Lf , cp = −1/Lf , and m(t) represents the sum of all uncertainties caused by the parameter variation and and load disturbances. m(t) dynamic is assumed to be bounded (m(t) < , is a positive constant). The current tracking error is defined as eg = io − icmd . The threeorder dynamic sliding surface sg (t) is defined as follows:
sg (t) = k1
eg ()d + eg (t) + k2 e˙ g (t),
(12)
0
s˙ g (t) = k1 eg (t) + e˙ g (t) + k2 e¨ g (t),
where x(t) = vo , u(t) = vcon , dp = − Rf /Lf , fp = −1/Cf Lf , gp = KPWM /Cf Lf , and n(t) represents the sum of all the uncertainties caused by the parameter variation and dynamic and load disturbances. n(t) is assumed bounded (n(t) < , is a positive constant). A voltage tracking error is defined as e = vo − vcmd . A three-order dynamic sliding surface s(t) is defined as follows: t
s(t) = k3
t
˙ e()d + k4 e(t) + e(t),
where io is the output current, icmd is the reference voltage command, and k1 and k2 are non-zero positive constants. The sliding-mode current inner loop is composed of three parts: the equivalent model controller, switching controller, and adaptive observation. The equivalent model controller functions as a specifier of the desired performance based on the inverter model. The output voltage is ugtr . The switching controller functions as a suppressor of the uncertainty and unpredictable perturbation such that the equivalent model controller performance can be exactly ensured. The output voltage is ugsw . The objective of the adaptive observation is to alleviate the chattering phenomenon, which is inevitable in the sliding-mode control method. The observation adaptively chooses the control gain (ˆ ) by estimating the upper bound of the uncertainties. According to Eqs. (11)–(13), the control law in the gridconnected mode is designed as follows: u = ugtr + ugsw ,
(14)
¨ g + ap icmd − i˙ cmd ], ugtr = −b−1 p [cp vs + (ap + k1 )eg + k2 e
(15)
ˆ (t)sgn(sg (t)), ugsw = −b−1 p
(16)
(19)
0
˙ ˙ s(t) = k3 e(t) + k4 e(t) + e¨ (t), (13)
(18)
(20)
where vo is the output voltage, vcmd is the reference voltage command, and k3 and k4 are the non-zero positive constants. The build-up of the proposed control scheme in every phase is the same as that of the sliding-mode current inner loop (Fig. 6(b)). The output voltages of the equivalent model controller and switching controller are utr and usw . ˆ is the control gain of the adaptive observation. According to the dynamic model, the control law is designed as follows: u = utr + usw , utr =
−gp−1 [k3 e + k4 e˙ + dp v˙ o
(21) + fp vo − v¨ cmd ],
(22)
usw = −gp−1 (t)sgn(s(t)), ˆ
(23)
(t) ˆ˙ =
(24)
s(t) l
,
where l is a positive constant. The stability analysis process is shown in Appendix A.2. 3.3. Virtual resistive output-impedance loop
ˆ˙ (t) = sg (t) /g ,
(17)
where g is a positive constant. The analysis process is shown in Appendix A.1 to prove the control law stability.
In the islanding mode, the droop control and power sharing accuracy rely on the impedance amplitude and angle. However, the accurate value of the equivalent output impedance is hard to measure or calculate because of the existence of the line impedance and the output impedance differences between the inverters. In this situation, the method that adds the virtual impedance loop into the control action is considered, and the analytical method in Ref. [22] is adopted.
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Fig. 6. Block diagram of the inner loop based on the adaptive sliding-mode control (a) in the grid-connected mode and (b) islanding mode.
Table 1 Details of the 3 four-wire inverter. Item
Symbol
Normal value
Filter inductor Resistance of the inductor Filter capacitor Nominal power Nominal voltage Nominal frequency Integration constant Derivative constant Integration constant Proportion constant
Lf Rf Cf S
4 mH 0.05 10 F 10 KVA 220 V 50 Hz 1000 1 9e9 1.4e5
vo fo k1 k2 k3 k4
First, the inverter with the sliding-mode inner closed-loop is analyzed. By comparing Eqs. (5) and (21), Eq. (25) is obtained as follows: ˙ k3 e(t) + k4 e(t) + e¨ (t) +
Rf io i˙ o + =− Cf Lf Cf
t
s() 0
d . l
(25)
By using Laplace transformation, Eq. (25) is expressed as follows:
The input reference voltage of the inner loop is then rewritten as follows:
vo = vcmd − Zo (s) · io (Lf s + Rf )s3 Zo (s) = · 5 Lf Cf s + k2 s4 + k1 s3 + k1 s2 + k2 s + 1
Fig. 7. Bode diagram of the output impedance with the Rd variation (from zero to one).
vo = (v∗cmd − Rd · io ) − Zo (s) · io (26)
where Zo (s) is the output-impedance transfer function. The detail parameters of the 3 four-wire inverter are listed in Table 1. The Bode diagram of the output impedance is shown in Fig. 7 (Rd = 0). The output impedance value has comparable resistive and inductive terms. For example, at a power frequency of 50 Hz, the output impedance is about −30 dB and 8◦ . At 150 Hz, the output impedance is about −30 dB and 15◦ . The equivalent output impedance obviously does not meet the requirements of the parallel operation for microgrid inverters [22,23]. The method that adds the virtual resistive impedance loop to the control structure is then used. The equivalent output impedance of the inverter is changed and fixed by proportionally dropping the output voltage reference v∗cmd to the output current (Fig. 5).
= v∗cmd − Zo ∗ (s) · io
,
(27)
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. The influence of Rd on the output impedance is shown in Fig. 7. The output impedance becomes progressively resistive at the 50, 150, 250, and 350 Hz frequencies with increasing Rd value. At the same time, the magnitudes of the output impedance at such frequencies tend to be 20 lgRd. Therefore, the requirements for parallel operation are met with the proper Rd value design. 3.4. Outer power sharing control loop The equivalent output impedance of the inverter affects the power sharing accuracy and determines the P/Q droop control
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strategy. With the increasing of Rd , the original output impedance and line impedance of the inverter can be neglected, and the equivalent output impedance must be able to meet the requirements for parallel operation. However, excessive value of Rd would make the voltage reference dropping sharply, and lead to a large system steady-state error. So there is an inherent tradeoff between the virtual impedance loop and output-voltage regulation. When the line impedance of the inverter is large and cannot be neglected, the conventional droop control method in Eq. (10) may cause adverse effects on control performance. In order to improve the stability of the controller and reduce the adverse effect on power sharing, we modified the conventional droop control method as follows: Ei = Ei∗ + m(Pi ∗ − Pi ) − nd (Qi∗ − Qi ),
(28)
ωi = ωi∗ − n(Qi∗ − Qi ) + md (Pi ∗ − Pi ).
(29)
In the islanding mode, m and n play a decisive role on power sharing because of the existence of the virtual resistive outputimpedance loop. Coefficients md and nd reduce the circulating current and improve the power sharing accuracy among inverters. According to the conventional droop control method, m and n values are obtained as m = E/Pmax and n = ω/Qmax , respectively. Pmax and Qmax are the maximum output active and reactive powers of the inverter, respectively. ω and E are the maximum angular frequency and the amplitude deviations of the output voltage. From Eq. (6), the output active power (Pi ) of the inverter is a function based on the voltage amplitude Ei and the phase angle deviation i . To differentiate Ei and i in Eq. (6), the following formula is used:
⎧ 1 ∂Pi ⎪ ⎪ ⎨ ∂Ei = Zi U cos(i − i ) . ⎪ ∂Pi 1 ⎪ = UEi sin(i − i ) ⎩ Zi ∂i
Parameters
Inverter 1
Inverter 2
Units
Lf Rf Cf S Zline m n Rd ωc md nd
4 0.05 10 10 0.04 + j0.05 3.11e-4 2e-4 0.4 10 5e-5 7e-6
4 0.04 10 5 0.05 + .01 6.22e-4 4e-4 0.8 10 5e-5 7e-6
mH F KVA V/W rad/s/Var rad/s V s/W rad/Var
impedance, Eqs. (35) and (36) are the conventional droop control methods. To investigate the stability and transient response of the system, the small-signal analysis is used [24]. Considering the effect of the LPFs, the small-signal dynamics of the active and reactive powers (Eqs. (6) and (7)) can be written as Pˆ i =
ωc U ˆ i) · (cos i · Eˆ i − Ei sin i · s + ωc Ri
ωc U ˆ i) · (sin i · Eˆ i + Ei cos i · Qˆ i = − s + ωc Ri
m m cot(i − i ). = Ei tan(i − i )
nEi Xline ˆ · Qi Ri
mXline ω ˆ i = nQˆ i − · Pˆ i Ei Ri
m Xline · , Ei Ri
(31)
s3 + As2 + Bs + C = 0
(41)
(32)
(33)
where Xline is the line inductive impedance and Ri is the output resistive impedance, which involves the line resistive Rline and equivalent output resistive of the inverter Rd . From Eq. (7), the relationship between nd and n can be similarly derived as follows: nd ≈ nEi ·
Xline . Ri
(34)
Therefore, the modified droop control method, which is applicable to the multi-inverters in parallel in the islanding mode, is obtained as follows: Xline ∗ (Qi − Qi ), Ri
(35)
m Xline ∗ · (Pi − Pi ). Ei Ri
(36)
Ei = Ei∗ + m(Pi ∗ − Pi ) − nEi · ωi = ωi∗ − n(Qi∗ − Qi ) +
(39) (40)
In a practical parallel operation system where i is very small and i is more than i , Eq. (32) is simplified as follows: md ≈
(38)
where ωc /(s + ωc ) is the LPF, and ωc is cutoff frequency, while Eˆ i and ˆ i denotes the perturbed values of Ei and i . By differentiating Eqs. (35) and (36), the following dynamics can be obtained:
then, md =
(37)
Then, the characteristic equation of the close-loop system can be obtained as follows:
The relationship between md and m is obtained as follows: 1 1 md ∂Pi /∂Ei = · = , m Ei tan(i − i ) ∂Pi /∂i
Table 2 Inverter parameters for a parallel operation.
Eˆ i = −mPˆ i + (30)
197
If the line inductive impedance is neglected or the equivalent output impedance of the inverter is much larger than the line
Fig. 8. Experimental setup.
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Fig. 9. Simulation results in the grid-connected mode with the given power command variation, (a) active power (upper trace) and reactive power (lower traces) responses of the INV1, (b) active power (upper trace) and reactive power (lower traces) responses of INV2, and (c) output currents of INV1 (upper trace), INV2 (middle trace) and grid (lower traces).
Where
4. Performance evaluation
Uωc m cos i Uωc nEi Xline sin i A = 2ωc + + Ri R2 i
B = ωc2 + (nEi + mωc )
Uωc cos i Uωc Xline sin i + (nEi ωc − m) Ri R2 i
C = (cos i Ri + mU)
ωc2 nUEi Ri2
+ (Ri2
+ nUEi Xline )
mUωc2 Xline Ri4
Using the parameters listed in Table 2, the characteristic roots which involve a negative real root and a couple of conjugate imaginary roots can be fixed. According to the qualitative analysis of root locus, the stability of the modified droop method can be ensured, and the dynamic performance is affected by the line inductive impedance Xline . With the increase of Xline , the dynamic performance becomes poor. However, in the actual low-voltage microgrid, Xline is usually much smaller than the line impedance, so the dynamic performance of the modified droop method also can be ensured.
In this section, the performance of the proposed control structure is evaluated by using the simulation and experiment. The simulation test system is shown in Fig. 1. The test involves two DG interfaces. Both topologies adopt combined three-phase inverter circuits. The circuit and control parameters are given in Table 2. The experimental setup which involves two inverters is shown in Fig. 8(a). Only the single-phase full-bridge circuits are used in the inverters because of the particularity of the combined three-phase four-wire topology, and the design capacities are 5 and 2.5 kW. The composition of INV1 is shown in Fig. 8(b). Compared with the INV1, the INV2 has a simpler circuit structure due to the use of IPM (Intelligent Power Module). The three cases are then discussed in the following sections. Case 1.
Power variation in the grid-connected mode
The simulation performance in the grid-connected mode is shown in Fig. 9. The initial INV states are P1 = 10 kW, Q1 = 0 kVar, P2 = kW, and Q2 = 0 kVar, where P and Q represent the active and reactive powers, respectively. The variation commands of the given power are set at t = 0.1 s and t = 0.3 s. Fig. 9(a) and (b) shows the
Fig. 10. Experiment results (a) grid voltage (upper trace), INV1 current (middle trace) and INV2 current (lower trace) responses in the grid-connected mode with the given power command variation, and (b) load voltage (upper trace), load current (middle trace) and grid current (lower trace) responses during the transition.
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Fig. 11. Simulation results during the transition from the grid-connected mode to the islanding mode, (a) active power responses of INV1 (upper trace) and INV2 (lower trace), (b) output currents of INV1 and INV2, (c) load voltage, (d) load current, and (e) grid current.
active and the reactive power responses of the two inverters. At moment t = 0.1 s. The output active power of INV1 is changed from 10 kW to 5 kW, and the output reactive power of INV2 is changed from 0 kW to 3 kW. At t = 0.3 s, all the power commands recover to the initial states. The output currents of the INVs and grid current are shown in Fig. 9(c). The currents in the grid-connected mode stabilize within one circle and have no effect on each other. The experiment results are shown in Fig. 10(a). The initial powers are set as P1 = 3 kW, Q1 = 0 kVar, P2 = 2 kW, and Q2 = 0 kVar. The given power value of INV1 decreases to 1.5 kW. The simulation and experiment results show that the proposed sliding-mode current controller is effective for the microgrid inverters in the grid-connected mode. Furthermore, a dynamic performance is ensured. Case 2. Transition from the grid-connected mode to the islanding mode The control performance during the transition from the gridconnected mode to the islanding mode is discussed in this case. The simulation results are shown in Fig. 11. The given power values are P1 = 10 kW and P2 = 5 kW. The distributed load is 6 kW. The active power responses and output current wavelets of the two inverters are shown in Fig. 11(a) and (b). The load voltage and current wavelets are shown in Fig. 11(c) and (d). The grid current, which flows from the DGs to the utility, is shown in Fig. 11(e). Three characteristics are concluded: (1) The voltage maintains stability when the inverter operates either in the grid-connected mode, islanding mode, or transition from the grid-connected mode to the islanding mode; therefore, the seamless formation is obtained. (2) The dynamic response during the transition is small enough to ensure the robustness of the proposed control structure. (3) The load power is proportionally shared by the microgrid inverters to their ratings.
Fig. 12. Simulation results under unbalanced loads, (a) load voltage, (b) load current, (c) output current of INV1, and (d) output current of INV2.
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Fig. 13. 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, and (f) output current spectra of INV2.
The experiment results during the transition are shown in Fig. 10(b). The mode transition disturbances are effectively rejected because of the robust sliding-mode control performance. The load current remains the same with the small load voltage variation.
Case 3.
Unbalanced and nonlinear loads in the islanding mode
To test the robustness of the proposed control structure in rejecting unbalanced disturbances, the unbalanced loads are added into the microgrid in the time period 0.1 s < t < 0.3 s. The simulation performances of the load voltage and the current under unbalanced loads are shown in Fig. 12(a) and (b). The three-phase current amplitudes and phases are different, but the load voltage remains balanced. The three-phase voltages are balanced because of the special topology used in this paper. The output current waveforms of the INVs are shown in Fig. 12(c) and (d). The current amplitudes of INV1 and INV2 are proportional to their power ratings. Moreover, their current phases are the same, thus ensuring the power sharing accuracy under unbalanced loads. The uncontrolled three-phase rectifier circuit is used to test the robustness of the inverters under the nonlinear loading condition, and the load power is set to 10 kW. The simulation results are shown in Fig. 13, and the load voltage is shown in Fig. 13(a). The frequency spectra are analyzed in Fig. 13(c). The output voltage, which yields a total harmonic distortion (THD) of 1.21%, is regulated to reject the nonlinear load disturbances. The load and output currents of the INVs are shown in Fig. 13(b). The frequency spectra are correspondingly analyzed in Fig. 13(d)–(f). The THDs of these currents (35%, 34.95%, and 35.19%) are nearly the same. The experiment results of the microgrid inverters under the islanding mode are shown in Fig. 14. The load power of the singlephase uncontrolled rectifier circuit is set to 1 kW. Compared with the simulation results, the load voltage and output currents (Fig. 14) have high THDs. However, the power sharing accuracy has not changed. The inverters share not only the fundamental currents but also the harmonic currents, thus ensuring the power sharing accuracy under nonlinear loads. The proposed control structure is reliably used under different microgrid operation modes.
Fig. 14. Experiment results under the nonlinear loads: load voltage (upper trace), output currents (lower trace) of INV1 and INV2.
5. Conclusion This paper presents a new solution for microgrid inverters in terms of circuit topology and control structure. The combined three-phase four-wire inverter, which is composed of three singlephase full-bridge inverter circuits, is adopted. The control structure is divided into three main parts: the islanding detection unit, the grid-connected controller, and the islanding controller. The inverters automatically operate in the grid-connected or the islanding mode using the islanding detection unit. In the grid-connected mode, the controller is composed of the sliding-mode current inner loop and the constant power control outer loop. The inverter effectively rejects the grid voltage disturbances and the parameter uncertainties. Moreover, the inverter also provides constant currents to the grid. In the islanding mode, the controller is based on the sliding-mode voltage inner loop, the virtual resistive output-impedance loop, and the improved P/Q sharing control loop. The controller guarantees the parallel operation of the microgrid inverters with robustness performance and a good power sharing accuracy. The dynamic characteristics and the high disturbance rejection performance during the transition is ensured. The theoretical analysis and test results validate the proposed control strategy effectiveness.
Y. Liu et al. / Electric Power Systems Research 117 (2014) 192–201
Appendix A. A.1. Stability proof of the control law in the grid-connected mode The Lyapunov function candidate is defined as follows: Vg (sg , ˜ ) =
sg2 2
+
g ˜ 2 2
where ˜ = − ˆ . The Lyapunov function candidate derivative is expressed as follows: V˙ g (sg (t), ˜ (t)) = sg (t)s˙ g (t) + g ˜ (t)˜˙ (t) According to Eqs. (11)–(15): V˙ g (sg (t), ˜ (t)) = sg (t)[bp ugsw + m(t)] + g [ − ˆ (t)]˜˙ (t) = sg (t)[−ˆ (t)sgn(sg (t)) + m(t)]−
g [ − ˆ (t)] sg (t) /g
= sg (t)m(t) − sg (t)
≤ sg (t) [m(t) − ] ≤0 A.2. Stability proof of the control law in the grid-connected mode The Lyapunov function candidate is defined as follows: Vl (s, ) ˜ =
s2 ˜2 + l 2 2
where ˜ = − . ˆ The Lyapunov candidate function derivative is expressed as follows: ˙ V˙ l (s(t), (t)) ˜ = s(t)s(t) + l (t) ˜ (t). ˜˙ According to Eqs. (18)–(22): ˜ = s(t)[gp usw + n(t)] + l [ − (t)] ˆ (t) ˜˙ V˙ l (s(t), (t)) = s(t)[−(t)sgn(s(t)) ˆ + n(t)]−
s(t) /l ˆ l [ − (t)] ≤ s(t) [n(t) − ] = s(t)n(t) − s(t)
.
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Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
A control strategy for microgrid inverters based on adaptive three-order sliding mode and optimized droop controls Yancheng Liu, Qinjin Zhang ∗ , Chuan Wang, Ning Wang Marine Engineering College, Dalian Maritime University, Dalian 116026, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 9 April 2014 Received in revised form 12 August 2014 Accepted 25 August 2014 Available online 16 September 2014 Keywords: Microgrid inverter Distributed generation (DG) Adaptive three-order sliding-mode control Droop control Disturbance rejection Nonlinear load
a b s t r a c t Robust control and seamless formation are the two crucial problems that affect smart microgrids. This paper proposes a new solution for microgrid inverters in terms of circuit topology and control structure. The combined three-phase four-wire inverter, which is composed of three single-phase full-bridge circuits, is adopted in this study. The control structure is based on the inner adaptive three-order slidingmode closed-loop, the immediate virtual output-impedance loop, and the outer power control loop. Three significant contributions are obtained: (1) the microgrid inverters effectively reject both voltage and load disturbances with the adaptive sliding-mode controllers regardless of whether the inverters are operating in the grid-connected mode, islanding mode, or transition from the grid-connected mode to the islanding mode; (2) the virtual output impedance loop is applied to make a resistive equivalent output impedance of the inverters and to meet the requirements of the inverter parallel operation in the islanding mode; (3) the proposed droop method reduces the line inductive impedance effects and improves the power sharing accuracy by optimizing the droop coefficients. The theoretical analysis and test results validate the effectiveness of the proposed control scheme. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The installation of distributed generations (DGs) has been given progressive attention not only because of the increasing concern toward environmental emissions from centralized power plants but also because of economic and technical reasons [1–4]. However, the penetration of DGs in a power system is limited because of technical reasons, such as stability constraints. IEEE Standard 1547.4 has recently proposed a clustering concept that uses a microgrid as the building block of DG power systems [5–7]. The microgrid should be able to operate in both the grid connected and islanding modes. Furthermore, the transition between the two modes should be seamless. DGs are usually interfaced through power electronic converters to provide loads. A flexible operation and a robust control of DG interfaces are the major objectives of smart microgrid research [6,7]. The output performance and the robustness of microgrid inverters are mainly affected by the effectiveness of the control strategy used [8–13]. The control strategy is usually composed of the outer power control loop and the inner voltage closed-loop. In the past
∗ Corresponding author. Tel.: +86 13664290653. E-mail addresses: [email protected] (Y. Liu), [email protected] (Q. Zhang), [email protected] (C. Wang), [email protected] (N. Wang). http://dx.doi.org/10.1016/j.epsr.2014.08.021 0378-7796/© 2014 Elsevier B.V. All rights reserved.
decade, various closed-loop control techniques (e.g., proportionalresonant control [9], Lyapunov-function-based control [10], two degrees-of-freedom control [11], H∞ control [12], and fuzzy control [13]) have achieved a dynamic characteristic and disturbance rejection under different load types. However, most of these works are only suitable for the grid-connected mode or islanding mode. Some recent works have focused on the sliding-mode control method [14–17]. A robust sliding-mode controller is proposed in Ref. [14] to control the active and reactive powers of a double fed induction generator (DFIG) wind system without involving any synchronous coordinate transformation. However, the DFIG wind system usually operates in the grid-connected mode and stops when a fault occurs or the power quality in the utility decreases. Wai et al. [15] design the adaptive total sliding-mode controllers, which correspond to different operation modes for single-phase inverters. However, the parallel operation has not been considered. Mohamed et al. [16] present a direct-voltage control strategy for a microgrid converter on the basis of the sliding-mode dynamic controller. The method can realize a normal operation in both the grid-connected and isolated modes. However, the output current performance in the grid-connected mode is unimpressive because of the indirect current control. In Ref. [17], a dual-loop controller is proposed for a voltage source inverter control. The inner loop is designed by using the sliding-mode control strategy, and the inner loop generates the pulse-width modulation voltage
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193
parallel operation. Section 3 presents the proposed control structure and the analysis. Section 4 shows the test results, which demonstrate the effectiveness and the applicability of the proposed control strategies. Section 5 concludes. 2. System descriptions
Fig. 1. A simplified microgrid system.
commands to regulate the inverter currents. Nevertheless, previous studies have not considered the chattering phenomenon of the sliding-mode control. The current study designs new closed-loop controllers that correspond to different operation modes on the basis of the adaptive three-order sliding-mode control. The closedloop adopts direct current control in the grid-connected mode, whereas the islanding mode adopts the direct-voltage control. The voltage and load disturbances are effectively rejected. Furthermore, the dynamic switching process from the grid-connected mode to the islanding mode is smooth. In order to solve the unbalanced problem, the combined three-phase four-wire inverter, which is composed of three single-phase full-bridge circuits, is adopted in the paper. Another problem for microgrid inverters in the islanding mode is the power sharing accuracy. The inverters should be able to proportionally share the distributed loads to their power ratings. The droop methods, which emulate the behavior of large power generators, are usually adopted to wirelessly perform functions [19–21]. The equivalent output impedance of the inverter should be resistive or inductive, which is the basic prerequisite for the application of the droop method. In the actual system, the output impedance is resistive–inductive and hard to measure or estimate. A possible solution is the addition of an inductor in the series with an inverter output. Nevertheless, this inductor is heavy and bulky and can cause an imbalance among the three phases. Hence, another method that places the virtual output impedance loop into the control structure is adopted [22,23]. This paper adopts a virtual resistive output impedance loop on this basis. Furthermore, an optimized droop controller that considers line impedance is designed. The power sharing accuracy and dynamic response can be effectively improved by controlling the current feedback gain and optimizing the droop coefficients. Section 2 describes the system, including the circuit topology, the system model, and the basic principle of the decentralized
A simplified microgrid system that consists of DGs, distributed loads, utility, voltage source inverters (VSIs) and AC bus is shown in Fig. 1. VSIs generally operate in the grid-connected mode, and the power is transmitted from the DGs into the utility. When a fault occurs or when the power quality in the utility worsens, the microgrid system disconnects from the utility by cutting off switch S1. The system then enters the intentional islanding mode. DGs and VSIs should be able to proportionally share the variable distributed loads to their power ratings and maintain the consistency of load voltage. If the utility recovers, the microgrid system reconnects to the utility. The dynamic models of the inverter in the grid connected and islanding modes differ according to the operation mode performance. 2.1. Inverter topology In the actual microgrid, the distributed loads are usually unbalanced, and sometimes the single-phase loads are dominant. So when it comes to the inverter design, the serious load imbalance problem should be considered firstly [18]. Then the VSI topology which is shown in Fig. 2 is adopted in this paper. The topology is composed of three single-phase full-bridge circuits (T1–T12), lowpass filters (Lf and Cf ), and an isolated transformer (T). Rf is the LC filter resistance per phase, and Zline is the line impedance between the inverter and bus. u and i are the output voltage and the current of the modular inverter circuit, respectively. vo and io are the output voltage and current of the low-pass filter. Subscripts a, b, and c represent the three phases. Furthermore, each phase in the topology can be controlled independently. 2.2. The model in the grid-connected mode Before analyzing the model, the following assumptions are made: (1) all switching devices are ideal and delay time is neglected; (2) the isolated transformer T is ideal, its turn ratio is 1:1 and its phase angle shift is not considered. Therefore, when the microgrid operates in the grid-connected mode, the dynamic equation of every phase in the VSI is represented as follows: dio 1 = (KPWM vcon − vs − io Rf − vsd ), Lf dt
Fig. 2. Topology of the three-phase four-wire inverter.
(1)
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is the sum of the output inductance and line inductance. The active and reactive powers of the inverter i are represented as follows: 1
Pi =
[(UEi cos i − U 2 ) cos i + UEi sin i sin i ], Zi
Qi =
[(UEi cos i − U 2 ) sin i − UEi sin i cos i ], Zi
1
(6)
(7)
where Zi is the impedance amplitude of the inverter i, Zi =
ri2 + Xi2 , and i is the impedance angle. Assuming that the impedance of the inverter is resistive (Zi = Ri ), the active and reactive powers are expressed as follows:
Fig. 3. VSI model (a) in the grid-connected mode and (b) islanding mode.
Pi =
1 (UEi cos i − U 2 ), Ri
Qi = −
UEi sin i . Ri
(8)
(9)
The conventional droop characteristics are used to conduct the parallel operation function:
ωi = ωi∗ − n(Qi∗ − Qi )
Fig. 4. Schematic of a microgrid with two DGs.
where KPWM is the equivalent parameter of the modular inverter circuit, vcon is the input control signal, KPWM vcon represents the output voltage of the full-bridge circuit, vs is the grid voltage, and vsd emulates the grid voltage disturbance. Eq. (1) can be represented as follows: Rf KPWM 1 1 i˙ o = vcon − vs − io − vsd . Lf Lf Lf Lf
(2)
The dynamic model of the VSI in the grid-connected mode is shown in Fig. 3(a). Laplace transformation is used in this model. 2.3. The model in the islanding mode When the microgrid operates in the islanding mode, the dynamic equation of every phase in the VSI is represented as follows: diLf dt
=
1 (KPWM vcon − vo − iLf Rf ), Lf
dvo 1 = (i − io ), Cf Lf dt
(3)
(4)
where vo is the output voltage of the L and C filters, and iLf is the inductance current. Eqs. (3) and (4) are then represented as follows:
v¨ o =
Rf Lf
v˙ o −
Rf 1 K 1 vo + PWM vcon − i˙ o − io . Cf Lf Cf Lf Cf Cf Lf
Ei = Ei∗ + m(Pi ∗ − Pi )
(5)
The dynamic model of the VSI in the islanding mode is shown in Fig. 3(b). Laplace transformation is used in this model. 2.4. Power sharing control The schematic of a microgrid with two DGs is shown in Fig. 4. U ∠ 0 is the AC bus voltage, E1 ∠ 1 and E2 ∠ 2 are the output voltages of the two inverters, is the phase angle difference between the output voltage and bus voltage. ri emulates the sum of the output resistance and line resistance of the inverter i (i = 1 or 2), and Xi
,
(10)
where ωi∗ and Ei∗ are nominal angular frequency and inverter voltage, respectively. m and n are droop coefficients. In the actual system, the equivalent output impedance of the inverter is hard to measure or calculate. In the low-voltage microgrid, the line impedance is usually resistive–inductive and has a value that is proportional to the length of the line. Under this circumstance, the resistance or the induction should not be neglected. Hence, the conventional droop method has a few limitations.
3. Proposed control scheme A new control structure for the three-phase four-wire inverter, which can operate in both the grid connected and the islanding modes, is proposed. The proposed control structure is shown in Fig. 5. The structure consists of six main parts: (1) the sliding-mode current inner loop; (2) sliding-mode voltage inner loop; (3) virtual output impedance medium loop; (4) constant power control outer loop; (5) P/Q sharing control outer loop; (6) islanding detection and mode selection. The microgrid inverter automatically operates in the grid connected or islanding mode by using the islanding detection and mode selection unit. In the grid-connected mode, the inverter is controlled by the sliding-mode current inner loop and constant power control outer loop. The inverter effectively rejects the grid voltage disturbances and parameter uncertainties. Moreover, the inverter provides a constant current to the grid. In the islanding mode, the inverter is controlled by the sliding-mode voltage inner loop, virtual output impedance medium loop, and P/Q sharing control outer loop. The controller guarantees the parallel operation of the microgrid inverters with robust performance and good power sharing accuracy. vs is the measure value of the grid voltage, and vo and io are the output voltage and the inverter current, respectively (Fig. 5). P* and Q* are the given values of the active and the reactive powers, respectively. The inner closed-loop and virtual output impedance medium loop are independently controlled for each phase, whereas the outer power loop is calculated for all lumped phases because of the particularity of the inverter topology adopted in this paper.
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195
Fig. 5. Proposed inverter control structure.
3.1. The inner closed-loop in the grid-connected mode
3.2. Inner closed-loop in the islanding mode
The main objective of the inner closed-loop in the gridconnected mode is to maximize the voltage disturbance rejection performance and obtain a good current tracking performance. From Eq. (2), the state equation of every phase is represented as follows:
The main objective of the inner closed-loop in islanding mode is to maximize the load disturbance rejection performance and obtain a good voltage tracking performance. From Eq. (5), the state equation of every phase is represented as follows:
x˙ g (t) = ap xg (t) + bp u(t) + cp z(t) + m(t)
˙ + fp x(t) + gp u(t) + n(t), x¨ (t) = dp x(t)
(11)
where xg (t) = io , u(t) = vcon , z(t) = vs , ap = − Rf /Lf , bp = KPWM /Lf , cp = −1/Lf , and m(t) represents the sum of all uncertainties caused by the parameter variation and and load disturbances. m(t) dynamic is assumed to be bounded (m(t) < , is a positive constant). The current tracking error is defined as eg = io − icmd . The threeorder dynamic sliding surface sg (t) is defined as follows:
sg (t) = k1
eg ()d + eg (t) + k2 e˙ g (t),
(12)
0
s˙ g (t) = k1 eg (t) + e˙ g (t) + k2 e¨ g (t),
where x(t) = vo , u(t) = vcon , dp = − Rf /Lf , fp = −1/Cf Lf , gp = KPWM /Cf Lf , and n(t) represents the sum of all the uncertainties caused by the parameter variation and dynamic and load disturbances. n(t) is assumed bounded (n(t) < , is a positive constant). A voltage tracking error is defined as e = vo − vcmd . A three-order dynamic sliding surface s(t) is defined as follows: t
s(t) = k3
t
˙ e()d + k4 e(t) + e(t),
where io is the output current, icmd is the reference voltage command, and k1 and k2 are non-zero positive constants. The sliding-mode current inner loop is composed of three parts: the equivalent model controller, switching controller, and adaptive observation. The equivalent model controller functions as a specifier of the desired performance based on the inverter model. The output voltage is ugtr . The switching controller functions as a suppressor of the uncertainty and unpredictable perturbation such that the equivalent model controller performance can be exactly ensured. The output voltage is ugsw . The objective of the adaptive observation is to alleviate the chattering phenomenon, which is inevitable in the sliding-mode control method. The observation adaptively chooses the control gain (ˆ ) by estimating the upper bound of the uncertainties. According to Eqs. (11)–(13), the control law in the gridconnected mode is designed as follows: u = ugtr + ugsw ,
(14)
¨ g + ap icmd − i˙ cmd ], ugtr = −b−1 p [cp vs + (ap + k1 )eg + k2 e
(15)
ˆ (t)sgn(sg (t)), ugsw = −b−1 p
(16)
(19)
0
˙ ˙ s(t) = k3 e(t) + k4 e(t) + e¨ (t), (13)
(18)
(20)
where vo is the output voltage, vcmd is the reference voltage command, and k3 and k4 are the non-zero positive constants. The build-up of the proposed control scheme in every phase is the same as that of the sliding-mode current inner loop (Fig. 6(b)). The output voltages of the equivalent model controller and switching controller are utr and usw . ˆ is the control gain of the adaptive observation. According to the dynamic model, the control law is designed as follows: u = utr + usw , utr =
−gp−1 [k3 e + k4 e˙ + dp v˙ o
(21) + fp vo − v¨ cmd ],
(22)
usw = −gp−1 (t)sgn(s(t)), ˆ
(23)
(t) ˆ˙ =
(24)
s(t) l
,
where l is a positive constant. The stability analysis process is shown in Appendix A.2. 3.3. Virtual resistive output-impedance loop
ˆ˙ (t) = sg (t) /g ,
(17)
where g is a positive constant. The analysis process is shown in Appendix A.1 to prove the control law stability.
In the islanding mode, the droop control and power sharing accuracy rely on the impedance amplitude and angle. However, the accurate value of the equivalent output impedance is hard to measure or calculate because of the existence of the line impedance and the output impedance differences between the inverters. In this situation, the method that adds the virtual impedance loop into the control action is considered, and the analytical method in Ref. [22] is adopted.
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Y. Liu et al. / Electric Power Systems Research 117 (2014) 192–201
Fig. 6. Block diagram of the inner loop based on the adaptive sliding-mode control (a) in the grid-connected mode and (b) islanding mode.
Table 1 Details of the 3 four-wire inverter. Item
Symbol
Normal value
Filter inductor Resistance of the inductor Filter capacitor Nominal power Nominal voltage Nominal frequency Integration constant Derivative constant Integration constant Proportion constant
Lf Rf Cf S
4 mH 0.05 10 F 10 KVA 220 V 50 Hz 1000 1 9e9 1.4e5
vo fo k1 k2 k3 k4
First, the inverter with the sliding-mode inner closed-loop is analyzed. By comparing Eqs. (5) and (21), Eq. (25) is obtained as follows: ˙ k3 e(t) + k4 e(t) + e¨ (t) +
Rf io i˙ o + =− Cf Lf Cf
t
s() 0
d . l
(25)
By using Laplace transformation, Eq. (25) is expressed as follows:
The input reference voltage of the inner loop is then rewritten as follows:
vo = vcmd − Zo (s) · io (Lf s + Rf )s3 Zo (s) = · 5 Lf Cf s + k2 s4 + k1 s3 + k1 s2 + k2 s + 1
Fig. 7. Bode diagram of the output impedance with the Rd variation (from zero to one).
vo = (v∗cmd − Rd · io ) − Zo (s) · io (26)
where Zo (s) is the output-impedance transfer function. The detail parameters of the 3 four-wire inverter are listed in Table 1. The Bode diagram of the output impedance is shown in Fig. 7 (Rd = 0). The output impedance value has comparable resistive and inductive terms. For example, at a power frequency of 50 Hz, the output impedance is about −30 dB and 8◦ . At 150 Hz, the output impedance is about −30 dB and 15◦ . The equivalent output impedance obviously does not meet the requirements of the parallel operation for microgrid inverters [22,23]. The method that adds the virtual resistive impedance loop to the control structure is then used. The equivalent output impedance of the inverter is changed and fixed by proportionally dropping the output voltage reference v∗cmd to the output current (Fig. 5).
= v∗cmd − Zo ∗ (s) · io
,
(27)
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. The influence of Rd on the output impedance is shown in Fig. 7. The output impedance becomes progressively resistive at the 50, 150, 250, and 350 Hz frequencies with increasing Rd value. At the same time, the magnitudes of the output impedance at such frequencies tend to be 20 lgRd. Therefore, the requirements for parallel operation are met with the proper Rd value design. 3.4. Outer power sharing control loop The equivalent output impedance of the inverter affects the power sharing accuracy and determines the P/Q droop control
Y. Liu et al. / Electric Power Systems Research 117 (2014) 192–201
strategy. With the increasing of Rd , the original output impedance and line impedance of the inverter can be neglected, and the equivalent output impedance must be able to meet the requirements for parallel operation. However, excessive value of Rd would make the voltage reference dropping sharply, and lead to a large system steady-state error. So there is an inherent tradeoff between the virtual impedance loop and output-voltage regulation. When the line impedance of the inverter is large and cannot be neglected, the conventional droop control method in Eq. (10) may cause adverse effects on control performance. In order to improve the stability of the controller and reduce the adverse effect on power sharing, we modified the conventional droop control method as follows: Ei = Ei∗ + m(Pi ∗ − Pi ) − nd (Qi∗ − Qi ),
(28)
ωi = ωi∗ − n(Qi∗ − Qi ) + md (Pi ∗ − Pi ).
(29)
In the islanding mode, m and n play a decisive role on power sharing because of the existence of the virtual resistive outputimpedance loop. Coefficients md and nd reduce the circulating current and improve the power sharing accuracy among inverters. According to the conventional droop control method, m and n values are obtained as m = E/Pmax and n = ω/Qmax , respectively. Pmax and Qmax are the maximum output active and reactive powers of the inverter, respectively. ω and E are the maximum angular frequency and the amplitude deviations of the output voltage. From Eq. (6), the output active power (Pi ) of the inverter is a function based on the voltage amplitude Ei and the phase angle deviation i . To differentiate Ei and i in Eq. (6), the following formula is used:
⎧ 1 ∂Pi ⎪ ⎪ ⎨ ∂Ei = Zi U cos(i − i ) . ⎪ ∂Pi 1 ⎪ = UEi sin(i − i ) ⎩ Zi ∂i
Parameters
Inverter 1
Inverter 2
Units
Lf Rf Cf S Zline m n Rd ωc md nd
4 0.05 10 10 0.04 + j0.05 3.11e-4 2e-4 0.4 10 5e-5 7e-6
4 0.04 10 5 0.05 + .01 6.22e-4 4e-4 0.8 10 5e-5 7e-6
mH F KVA V/W rad/s/Var rad/s V s/W rad/Var
impedance, Eqs. (35) and (36) are the conventional droop control methods. To investigate the stability and transient response of the system, the small-signal analysis is used [24]. Considering the effect of the LPFs, the small-signal dynamics of the active and reactive powers (Eqs. (6) and (7)) can be written as Pˆ i =
ωc U ˆ i) · (cos i · Eˆ i − Ei sin i · s + ωc Ri
ωc U ˆ i) · (sin i · Eˆ i + Ei cos i · Qˆ i = − s + ωc Ri
m m cot(i − i ). = Ei tan(i − i )
nEi Xline ˆ · Qi Ri
mXline ω ˆ i = nQˆ i − · Pˆ i Ei Ri
m Xline · , Ei Ri
(31)
s3 + As2 + Bs + C = 0
(41)
(32)
(33)
where Xline is the line inductive impedance and Ri is the output resistive impedance, which involves the line resistive Rline and equivalent output resistive of the inverter Rd . From Eq. (7), the relationship between nd and n can be similarly derived as follows: nd ≈ nEi ·
Xline . Ri
(34)
Therefore, the modified droop control method, which is applicable to the multi-inverters in parallel in the islanding mode, is obtained as follows: Xline ∗ (Qi − Qi ), Ri
(35)
m Xline ∗ · (Pi − Pi ). Ei Ri
(36)
Ei = Ei∗ + m(Pi ∗ − Pi ) − nEi · ωi = ωi∗ − n(Qi∗ − Qi ) +
(39) (40)
In a practical parallel operation system where i is very small and i is more than i , Eq. (32) is simplified as follows: md ≈
(38)
where ωc /(s + ωc ) is the LPF, and ωc is cutoff frequency, while Eˆ i and ˆ i denotes the perturbed values of Ei and i . By differentiating Eqs. (35) and (36), the following dynamics can be obtained:
then, md =
(37)
Then, the characteristic equation of the close-loop system can be obtained as follows:
The relationship between md and m is obtained as follows: 1 1 md ∂Pi /∂Ei = · = , m Ei tan(i − i ) ∂Pi /∂i
Table 2 Inverter parameters for a parallel operation.
Eˆ i = −mPˆ i + (30)
197
If the line inductive impedance is neglected or the equivalent output impedance of the inverter is much larger than the line
Fig. 8. Experimental setup.
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Fig. 9. Simulation results in the grid-connected mode with the given power command variation, (a) active power (upper trace) and reactive power (lower traces) responses of the INV1, (b) active power (upper trace) and reactive power (lower traces) responses of INV2, and (c) output currents of INV1 (upper trace), INV2 (middle trace) and grid (lower traces).
Where
4. Performance evaluation
Uωc m cos i Uωc nEi Xline sin i A = 2ωc + + Ri R2 i
B = ωc2 + (nEi + mωc )
Uωc cos i Uωc Xline sin i + (nEi ωc − m) Ri R2 i
C = (cos i Ri + mU)
ωc2 nUEi Ri2
+ (Ri2
+ nUEi Xline )
mUωc2 Xline Ri4
Using the parameters listed in Table 2, the characteristic roots which involve a negative real root and a couple of conjugate imaginary roots can be fixed. According to the qualitative analysis of root locus, the stability of the modified droop method can be ensured, and the dynamic performance is affected by the line inductive impedance Xline . With the increase of Xline , the dynamic performance becomes poor. However, in the actual low-voltage microgrid, Xline is usually much smaller than the line impedance, so the dynamic performance of the modified droop method also can be ensured.
In this section, the performance of the proposed control structure is evaluated by using the simulation and experiment. The simulation test system is shown in Fig. 1. The test involves two DG interfaces. Both topologies adopt combined three-phase inverter circuits. The circuit and control parameters are given in Table 2. The experimental setup which involves two inverters is shown in Fig. 8(a). Only the single-phase full-bridge circuits are used in the inverters because of the particularity of the combined three-phase four-wire topology, and the design capacities are 5 and 2.5 kW. The composition of INV1 is shown in Fig. 8(b). Compared with the INV1, the INV2 has a simpler circuit structure due to the use of IPM (Intelligent Power Module). The three cases are then discussed in the following sections. Case 1.
Power variation in the grid-connected mode
The simulation performance in the grid-connected mode is shown in Fig. 9. The initial INV states are P1 = 10 kW, Q1 = 0 kVar, P2 = kW, and Q2 = 0 kVar, where P and Q represent the active and reactive powers, respectively. The variation commands of the given power are set at t = 0.1 s and t = 0.3 s. Fig. 9(a) and (b) shows the
Fig. 10. Experiment results (a) grid voltage (upper trace), INV1 current (middle trace) and INV2 current (lower trace) responses in the grid-connected mode with the given power command variation, and (b) load voltage (upper trace), load current (middle trace) and grid current (lower trace) responses during the transition.
Y. Liu et al. / Electric Power Systems Research 117 (2014) 192–201
199
Fig. 11. Simulation results during the transition from the grid-connected mode to the islanding mode, (a) active power responses of INV1 (upper trace) and INV2 (lower trace), (b) output currents of INV1 and INV2, (c) load voltage, (d) load current, and (e) grid current.
active and the reactive power responses of the two inverters. At moment t = 0.1 s. The output active power of INV1 is changed from 10 kW to 5 kW, and the output reactive power of INV2 is changed from 0 kW to 3 kW. At t = 0.3 s, all the power commands recover to the initial states. The output currents of the INVs and grid current are shown in Fig. 9(c). The currents in the grid-connected mode stabilize within one circle and have no effect on each other. The experiment results are shown in Fig. 10(a). The initial powers are set as P1 = 3 kW, Q1 = 0 kVar, P2 = 2 kW, and Q2 = 0 kVar. The given power value of INV1 decreases to 1.5 kW. The simulation and experiment results show that the proposed sliding-mode current controller is effective for the microgrid inverters in the grid-connected mode. Furthermore, a dynamic performance is ensured. Case 2. Transition from the grid-connected mode to the islanding mode The control performance during the transition from the gridconnected mode to the islanding mode is discussed in this case. The simulation results are shown in Fig. 11. The given power values are P1 = 10 kW and P2 = 5 kW. The distributed load is 6 kW. The active power responses and output current wavelets of the two inverters are shown in Fig. 11(a) and (b). The load voltage and current wavelets are shown in Fig. 11(c) and (d). The grid current, which flows from the DGs to the utility, is shown in Fig. 11(e). Three characteristics are concluded: (1) The voltage maintains stability when the inverter operates either in the grid-connected mode, islanding mode, or transition from the grid-connected mode to the islanding mode; therefore, the seamless formation is obtained. (2) The dynamic response during the transition is small enough to ensure the robustness of the proposed control structure. (3) The load power is proportionally shared by the microgrid inverters to their ratings.
Fig. 12. Simulation results under unbalanced loads, (a) load voltage, (b) load current, (c) output current of INV1, and (d) output current of INV2.
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Fig. 13. 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, and (f) output current spectra of INV2.
The experiment results during the transition are shown in Fig. 10(b). The mode transition disturbances are effectively rejected because of the robust sliding-mode control performance. The load current remains the same with the small load voltage variation.
Case 3.
Unbalanced and nonlinear loads in the islanding mode
To test the robustness of the proposed control structure in rejecting unbalanced disturbances, the unbalanced loads are added into the microgrid in the time period 0.1 s < t < 0.3 s. The simulation performances of the load voltage and the current under unbalanced loads are shown in Fig. 12(a) and (b). The three-phase current amplitudes and phases are different, but the load voltage remains balanced. The three-phase voltages are balanced because of the special topology used in this paper. The output current waveforms of the INVs are shown in Fig. 12(c) and (d). The current amplitudes of INV1 and INV2 are proportional to their power ratings. Moreover, their current phases are the same, thus ensuring the power sharing accuracy under unbalanced loads. The uncontrolled three-phase rectifier circuit is used to test the robustness of the inverters under the nonlinear loading condition, and the load power is set to 10 kW. The simulation results are shown in Fig. 13, and the load voltage is shown in Fig. 13(a). The frequency spectra are analyzed in Fig. 13(c). The output voltage, which yields a total harmonic distortion (THD) of 1.21%, is regulated to reject the nonlinear load disturbances. The load and output currents of the INVs are shown in Fig. 13(b). The frequency spectra are correspondingly analyzed in Fig. 13(d)–(f). The THDs of these currents (35%, 34.95%, and 35.19%) are nearly the same. The experiment results of the microgrid inverters under the islanding mode are shown in Fig. 14. The load power of the singlephase uncontrolled rectifier circuit is set to 1 kW. Compared with the simulation results, the load voltage and output currents (Fig. 14) have high THDs. However, the power sharing accuracy has not changed. The inverters share not only the fundamental currents but also the harmonic currents, thus ensuring the power sharing accuracy under nonlinear loads. The proposed control structure is reliably used under different microgrid operation modes.
Fig. 14. Experiment results under the nonlinear loads: load voltage (upper trace), output currents (lower trace) of INV1 and INV2.
5. Conclusion This paper presents a new solution for microgrid inverters in terms of circuit topology and control structure. The combined three-phase four-wire inverter, which is composed of three singlephase full-bridge inverter circuits, is adopted. The control structure is divided into three main parts: the islanding detection unit, the grid-connected controller, and the islanding controller. The inverters automatically operate in the grid-connected or the islanding mode using the islanding detection unit. In the grid-connected mode, the controller is composed of the sliding-mode current inner loop and the constant power control outer loop. The inverter effectively rejects the grid voltage disturbances and the parameter uncertainties. Moreover, the inverter also provides constant currents to the grid. In the islanding mode, the controller is based on the sliding-mode voltage inner loop, the virtual resistive output-impedance loop, and the improved P/Q sharing control loop. The controller guarantees the parallel operation of the microgrid inverters with robustness performance and a good power sharing accuracy. The dynamic characteristics and the high disturbance rejection performance during the transition is ensured. The theoretical analysis and test results validate the proposed control strategy effectiveness.
Y. Liu et al. / Electric Power Systems Research 117 (2014) 192–201
Appendix A. A.1. Stability proof of the control law in the grid-connected mode The Lyapunov function candidate is defined as follows: Vg (sg , ˜ ) =
sg2 2
+
g ˜ 2 2
where ˜ = − ˆ . The Lyapunov function candidate derivative is expressed as follows: V˙ g (sg (t), ˜ (t)) = sg (t)s˙ g (t) + g ˜ (t)˜˙ (t) According to Eqs. (11)–(15): V˙ g (sg (t), ˜ (t)) = sg (t)[bp ugsw + m(t)] + g [ − ˆ (t)]˜˙ (t) = sg (t)[−ˆ (t)sgn(sg (t)) + m(t)]−
g [ − ˆ (t)] sg (t) /g
= sg (t)m(t) − sg (t)
≤ sg (t) [m(t) − ] ≤0 A.2. Stability proof of the control law in the grid-connected mode The Lyapunov function candidate is defined as follows: Vl (s, ) ˜ =
s2 ˜2 + l 2 2
where ˜ = − . ˆ The Lyapunov candidate function derivative is expressed as follows: ˙ V˙ l (s(t), (t)) ˜ = s(t)s(t) + l (t) ˜ (t). ˜˙ According to Eqs. (18)–(22): ˜ = s(t)[gp usw + n(t)] + l [ − (t)] ˆ (t) ˜˙ V˙ l (s(t), (t)) = s(t)[−(t)sgn(s(t)) ˆ + n(t)]−
s(t) /l ˆ l [ − (t)] ≤ s(t) [n(t) − ] = s(t)n(t) − s(t)
.
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