energies Article
Adaptive Higher-Order Sliding Mode Control for Islanding and Grid-Connected Operation of a Microgrid Yaozhen Han 1,2, *, Ronglin Ma 1 1 2 3
*
ID
and Jinghan Cui 2,3
School of Information Science and Electrical Engineering, Shandong Jiaotong University, Jinan 250357, China;
[email protected] State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China;
[email protected] Department of Chemical & Materials Engineering Faculty of Engineering, University of Alberta, Edmonton, AB T5J4P6, Canada Correspondence:
[email protected]; Tel.: +86-0531-8068-7920
Received: 3 May 2018; Accepted: 1 June 2018; Published: 5 June 2018
Abstract: Grid-connected and islanding operations of a microgrid are often influenced by system uncertainties, such as load parameter variations and unmodeled dynamics. This paper proposes a novel adaptive higher-order sliding mode (AHOSM) control strategy to enhance system robustness and handle an unknown uncertainty upper bounds problem. Firstly, microgrid models with uncertainties are established under islanding and grid-connected modes. Then, adaptive third-order sliding mode and adaptive second-order sliding mode control schemes are respectively designed for the two modes. Microgrid models’ descriptions are divided into nominal part and uncertain part, and higher-order sliding mode (HOSM) control problems are transformed into finite time stability problems. Again, a scheduled law is proposed to increase or decrease sliding mode control gain adaptively. Real higher-order sliding modes are established, and finite time stability is proven based on the Lyapunov method. In order to achieve smooth mode transformation, an islanding mode detection algorithm is also adopted. The proposed control strategy accomplishes voltage control and current control of islanding mode and grid-connected mode. Control voltages are continuous, and uncertainty upper bounds are not required. Furthermore, adjustable control gain can further whittle control chattering. Simulation experiments verify the validity and robustness of the proposed control scheme. Keywords: microgrid control; adaptive higher-order sliding mode; unknown uncertainty upper bounds; robustness
1. Introduction A microgrid, which mainly includes a direct current (DC) form, an alternate current (AC) form, and an AC-DC form, is a small power system consisting of a distributed microsource, energy storing device, energy conversion device, and a load and control protection device [1]. It can provide clean energy for a remote area, island or city community and achieve combined cooling, heating, and power [2,3]. Microgrids have become a new driving force for the development of distributed renewable energy sources [4]. Microgrid operations consist of a grid-connected mode and an islanding mode. In grid-connected mode, voltage and frequency references are provided by the main grid and each distributed generation unit (DGu) achieves active power and reactive power regulation via current control. In islanding mode, voltage and frequency reference are maintained by the main DGu of a microgrid [5]. Energies 2018, 11, 1459; doi:10.3390/en11061459
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In the form of DC distribution, distributed microsources of a DC microgrid are merged together and coordinately controlled. Study on DC microgrid control has procured plentiful and substantial progeny [6,7]. However, for the AC microgrid, microgrid voltage control under the two modes is one of the key technologies for achieving safe and stable operation [8,9]. Many control methods are applied in the microgrid operation [10]. Droop control is a common voltage control method under islanding mode [11,12]. However, conventional droop control has some drawbacks, such as slow transient response and high dependence of filter impedance [13]. Dynamic load is also not included in the control loop, of which rapid or large variation may result in instability of voltage or frequency. Moreover, this method is not robust for microgrid parameters perturbation and external disturbance and can only make the microgrid steadily operate around a small operating region, which severely restricts the popularization and application of microgrid technology [14]. Sliding mode control, viewed as a variable structure nonlinear control method, has many advantages, such as rapid response and insensitivity to parametric variation and disturbance. It is suitable for power system robust control. Recently, some attempts have been made to study microgrid sliding mode control under grid-connected mode or islanding mode and have achieved corresponding robustness with parametric variation and external disturbance [15–19]. However, this literature does not only have its own drawbacks, such as single mode operation and lack of mode transformation strategy but also has the common problem of first-order sliding mode, which is a notorious control chattering phenomenon. Chattering can severely shorten the service life of power electronic devices in a microgrid system. Higher-order sliding mode (HOSM), with higher sliding accuracy, is suitable for high relative degree systems and can further restrain chattering by means of increasing system relative degree and hiding discontinuous term under time derivative of control term [20,21]. HOSM control has been a hot topic and widely applied in power system control [22,23]. Some scholars have attempted to study HOSM control for microgrid operation. Cucuzzella et al. [24] designed a suboptimal second-order sliding mode control scheme for islanding operation of a master-slave microgrid. Yet, the system relative degree was also two, which signified that the output control voltage of the voltage source inverter (VSI) was discontinuous and that the inhibitory effect of chattering is poor. In [25], third-order sliding mode control for the microgrid inverter was accomplished based on a combined linear sliding surface. However, the system only had asymptotic stability. Finite time stability for islanding operation was achieved based a third-order sliding mode control scheme in [26]. However, the microgrid model (controlled object) was treated as a black box in this method, which led to conservative parameter choice. Though control action can be continuous by artificially increasing relative degree, conservative control parameters will undoubtedly aggravate chattering. Meanwhile, an islanding mode detection algorithm is not mentioned which may extend transient response and cannot comply with IEEE Std. 1547-2003 [27]. Additionally, uncertainty upper bounds of the microgrid are supposed to be known in all aforementioned studies. However, no matter what, it is difficult to factually estimate the modeling error of VSI or system parameters. Conservative parameter choice may increase control action and intensify chattering phenomenon. Thus, HOSM control combining with an adaptive control gain method can fit well with this situation. Control gain can be adaptively regulated according to the amplitude of uncertainty upper bounds; meanwhile real HOSM is established. Paper [28] proposes an adaptive suboptimal second-order sliding mode control method for grid-connected operation. The control gain can be regulated according to uncertainty variation and control chattering is greatly whittled. However, the known part of the system model is viewed as uncertainty which increasing the controller burden. In this paper, a novel AHOSM control strategy is proposed for islanding and grid-connected operations of a microgrid. Mathematic models of microgrids under islanding mode and grid-connected mode are established first. The HOSM control problem is then converted to a finite time stability problem under islanding mode and adaptive third-order sliding mode voltage controller is designed
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combining control law with adaptive sliding mode control law. Energies 2018, homogeneous 11, x FOR PEER REVIEW
In grid-connected mode, 3 of 18 an adaptive second-order sliding mode control scheme is proposed to track current demand. Microgrid control under islanding and control grid-connected modes are both based on adaptive sliding current demand. Microgrid under islanding and grid-connected modeshigher-order are both based on mode (AHOSM), whichsliding can conquer uncertainty upper bounds, whittleuncertainty control chattering, adaptive higher-order mode unknown (AHOSM), which can conquer unknown upper and achieve voltage or current regulation. Islanding event detection reconnection algorithm is bounds, whittle control chattering, and achieve voltage or currentand regulation. Islanding event also adopted achieve smooth mode is transformation. detection andto reconnection algorithm also adopted to achieve smooth mode transformation. The paper paperisisorganized organized as follows: Section 2 is microgrid modeling. Thestrategy controlincluding strategy as follows: Section 2 is microgrid modeling. The control including adaptive third-order sliding mode control for islanding mode, adaptive second-order adaptive third-order sliding mode control for islanding mode, adaptive second-order sliding mode sliding control for grid-connected and event an islanding event and reconnection control mode for grid-connected mode, and anmode, islanding detection anddetection reconnection algorithm is algorithm proposed in Section 3. Section 4 presents results. some simulation results. Finallyremarks some, proposed inisSection 3. Section 4 presents some simulation Finally some, concluding concluding are given in Section 5. are given inremarks Section 5. 2. Microgrid Microgrid Modeling Modeling 2. A single-line a master-slave microgrid with with nDGus [26,29][26,29] is shortly shown in Figurein1. A single-linediagram diagramofof a master-slave microgrid nDGus is shortly shown Each microsource is a generally renewable energy type andtype is represented by a DCby voltage source, Figure 1. Each microsource is a generally renewable energy and is represented a DC voltage which is connected with the main grid via VSI, RL filter, and PCC. Three-phase balanced RCL load is source, which is connected with the main grid via VSI, RL filter, and PCC. Three-phase balanced connected the microgrid. RCL load iswith connected with the microgrid. vt , abcM
I t , abcs
vabc
I t , abcM θ (t )
vt , abcS
θ (t )
θ (t )
θ (t ) θ (t ) θ (t )
Figure Figure 1. 1. Single-line Single-line diagram diagram of of aa master-slave master-slave microgrid. microgrid.
Voltage magnitude and frequency of PCC are determined by the main grid under Voltage magnitude and frequency of PCC are determined by the main grid under grid-connected grid-connected mode. Thus, the microgrid is stiff synchronous with the main grid, and the reference mode. Thus, the microgrid is stiff synchronous with the main grid, and the reference angle of Park vabc and VSI angle of Park transformation is provided by PLL. d-q voltage component of load voltagecurrent transformation is provided by PLL. d-q component of load v abc and VSI output it,abc are it,, abc Vq Ito I tq VqThen =0 active V defined as V , V I , I . PI control is adopted stabilize V = 0 to lock main grid phase. q tq q d td d td are defined as , , , . PI control is adopted to stabilize to lock output current powergrid and phase. reactiveThen power can be respectively represented P =be3/2V d Itd , Q = 3/2V d Itq . Thus, main active power and reactive powerascan respectively represented as each DGu executes current control to regulate active power and reactive power. P = 3 / 2Vd I td , Q = 3 / 2Vd I tq . Thus, each DGu executes current control to regulate active power and The microgrid turns to islanding mode when breaker SW 2 is open. Due to the power mismatching reactive betweenpower. the DGu and load, PCC voltage and frequency can acutely deviate from the nominal value The microgrid turns islanding mode when breaker 2 is open. Due to the power at this time. Thus, the DGutoshould control voltage to track loadSW voltage reference under islanding mismatching between the DGu and load, PCC voltage and frequency canitacutely the mode. Park angular transformation is provided by inner oscillator and here is set asdeviate nominalfrom angular nominal value at this time. Thus, the DGu should control voltage to track load voltage reference frequency ω0 = 2π f 0 . The islanding event detection algorithm should be designed to smoothly transfer under islanding mode. Parktoangular transformation provided by innerneeds oscillator here itphase is set from grid-connected mode islanding mode. Hence,isthe inner oscillator PLL and to provide ω = 2 π f 0 . The as nominal angular frequency islanding detection should be angle as the initial condition under0 islanding mode and PCCevent voltage must be algorithm synchronous with the designed smoothly transferis from grid-connected mode to islanding mode. Hence, the inner main grid to when the microgrid reconnected. oscillator needs PLLatomaster-slave provide phase angle as the initial condition under islandingcurrent mode control and PCC Figure 1 shows microgrid control scheme. All DGus implement to voltage synchronous with the main grid when the microgrid is reconnected. achieve must activebepower and reactive power regulation under grid-connected mode. The master DGu Figure 1 showscontrol a master-slave microgrid scheme. Alland DGus implement switches to voltage to maintain desiredcontrol microgrid voltage frequency, and current the slavecontrol DGus to achieve active power and power regulation still regulate power when thereactive islanding event happens.under grid-connected mode. The master DGu switches to voltage control to maintain desired microgrid voltage and frequency, and the slave DGus still regulate power when the islanding event happens. Considering the symmetrical balanced system shown in Figure 1, according to the Kirchhoff voltage and current law, the DGu governing equations under islanding mode is:
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Considering the symmetrical balanced system shown in Figure 1, according to the Kirchhoff voltage and current law, the DGu governing equations under islanding mode is: dv abc 1 it,abc = R v abc + i L,abc + C dt dit,abc + Rt it,abc + v abc vt,abc = Lt dt di v abc = L L,abc + Rl i L,abc dt
(1)
where, it,abc , v abc , i L,abc , vt,abc are the DGu output current, load current, inductive load L current and VSI output voltage. According to the Clark and Park transformation, every three phase variables in Formula (1) can be represented under a d-q rotating coordinate system: . 1 Vd + ω0 Vq + C1 Itd − C1 ILd V d = − RC . 1 V q = −ω0 Vd − RC Vq + C1 Itq − C1 ILq . I = − 1 V − Rt I + ω I + 1 V + 1 u 0 tq td Lt d Lt td Lt td Lt VSId . Rt 1 1 I tq = − Lt Vq − ω0 Itd − Lt Itq + Lt Vtq + L1t uVSIq . Rl 1 I Ld = L Vd − L ILd + ω0 ILq . I Lq = L1 Vq − ω0 ILd − RLl ILq
(2)
where, uVSId and uVSIq are matched uncertainties caused by VSI. x = [Vd , Vq , Itd , Itq , ILd , ILq ] is chosen as system state variable, u = [Vtd , Vtq ] is the control input and y I = [Vd , Vq ] is an output vector. The DGu state space model under islanding mode is:
.
1 x1 (t) + ω0 x2 (t) + C1 x3 (t) − C1 x5 (t) x1 (t) = − RC . 1 x2 (t) = −ω0 x1 (t) − RC x2 (t) + C1 x4 (t) − C1 x6 (t) . Rt 1 x3 (t) = − Lt x1 (t) − Lt x3 (t) + ω0 x4 (t) + L1t u1 (t) + . x4 (t) = − L1t x2 (t) − ω0 x3 (t) − RLtt x4 (t) + L1t u2 (t) + . x5 (t) = L1 x1 (t) − RLl x5 (t) + ω0 x6 (t) . x6 (t) = L1 x2 (t) − ω0 x5 (t) − RLl x6 (t) y Id = x1 (t) y Iq = x2 (t)
1 Lt uVSId 1 Lt uVSIq
(3)
Similarly, the DGu state space model under grid-connected mode is:
.
1 x1 (t) = − RC x1 (t) + ωx2 (t) + C1 x3 (t) − C1 x5 (t) − C1 x7 (t) . 1 x2 (t) = −ωx1 (t) − RC x2 (t) + C1 x4 (t) − C1 x6 (t) − C1 x8 (t) . Rt 1 x3 (t) = − Lt x1 (t) − Lt x3 (t) + ωx4 (t) + L1t u1 (t) + L1t uVSId . x4 (t) = − L1t x2 (t) − ωx3 (t) − RLtt x4 (t) + L1t u2 (t) + L1t uVSIq . x5 (t) = L1 x1 (t) − RLl x5 (t) + ωx6 (t) . x6 (t) = L1 x2 (t) − ωx5 (t) − RLl x6 (t) . x7 (t) = L1s x1 (t) − RLss x7 (t) + ωx8 (t) − L1s u gd (t) . x8 (t) = L1s x2 (t) − ωx7 (t) − RLss x8 (t) − L1s u gq (t) yGd = x3 (t) yGq = x4 (t)
(4)
where x = [Vd , Vq , Itd , Itq , ILd , ILq , Igd , Igq ] is the state vector, u = [Vtd , Vtq ] is the control input, yG = [ Itd , Itq ] is the output, Igd , Igq , u gd , u gq are the exchange current and d-q voltages of main grid, u gd , u gq are the constant rated nominal values.
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3. Control Strategy The AHOSM control scheme for islanding mode and grid-connected mode operation and the islanding event detection algorithm are presented in this section. 3.1. Adaptive Third-Order Sliding Mode Control for Islanding Operation of the Microgrid In islanding mode, the master DGu is switched to voltage control manner to maintain voltage stabilization of the microgrid system. Sliding mode variables are chosen as: (
s Id = VIdre f − Vd s Iq = VIqre f − Vq
(5)
The system relative degree is easily calculated as two and is artificially increased to three for whittling high frequency oscillation of control variables and alleviating the harmonic influence of the electric signal. Then: ... s Id
. Rt Rt 1 1 1 0 = − 2ω + x − − x ( t ) − x ( t ) + ω x ( t ) + u ( t ) 2 3 0 1 4 1 2 Lt Lt Lt Lt C RC RC . Rt 1 1 1 0 x1 + 2ω − x ( t ) − ω x ( t ) − x ( t ) + u ( t ) − − R21C2 + ω02 + L1t C + LC 2 0 3 Lt 4 Lt 2 C Lt 2ω0 1 2ω0 . Rt . Rt 1 1 1 1 . + RC2 + LC x5 + C x6 + Lt C uVSId − RC2 + Lt C Lt uVSId + C Lt uVSIq ... . + L1t C ud + V Idre f = f Id + ∆ f Id + ( g Id + ∆g Id )vd
(6)
. . 2ω0 1 Rt 1 1 where vd = ud , ∆ f Id = L1t C uVSId − RC + 2 Lt C Lt uVSId + C Lt uVSIq + ∆ f Idv , ∆ f Idv is the uncertainty including model error and parameter deviation of f Id , ∆g Id is the uncertain part of g Id . ... s Iq
. 2ω0 1 1 − L1t x1 (t) − RLtt x3 (t) + ω0 x4 (t) + L1t u1 (t) Lt C + LC x2 − C . . Rt Rt 1 1 1 0 0 + 2ω + − x ( t ) − ω x ( t ) − x ( t ) + u ( t ) + 2ω x + x5 2 0 3 2 4 1 2 L L L RC L C C RC t t t t 2ω0 1 Rt . Rt 1 1 1 . 1 − − RC + 2 − LC x 6 + Lt C uVSIq − C Lt uVSId + 2 Lt uVSIq L C RC t ... . +V Iqre f + L1t C uq
= − ω02 −
=
1
R2 C 2
+
(7)
f Iq + ∆ f Iq + ( g Iq + ∆g Iq )vq
. . Rt 1 1 0 1 where vq = uq , ∆ f Iq = L1t C uVSIq − 2ω u + + VSI 2 C Lt Lt C Lt uVSIq + ∆ f Iqv , ∆ f Iqv is the uncertainty d RC including model error and parameter deviation of f Iq , ∆g Iq is the uncertain part of g Iq . ∆ f Ii and ∆g Ii (i = d,q) are both bounded and the upper bounds are unknown in the microgrid −1 system and satisfy g Ii ∆g Ii ≤ 1 − aˆ Ii , ∆ f Ii ≤ fˆIi , 0 < aˆ Ii ≤ 1. fˆIi is unknown. Third-order sliding mode control with respect to s Ii is equivalently converted to a finite time stabilization problem: . z. 1i = z2i z2i = z3i z. = f + ∆ f + ( g + gˆ )v 3i Ii Ii Ii Ii i .
(8)
..
where z1i = s Ii , z2i = s Ii and z3i = s Ii . Considering state feedback control: vi = g Ii −1 (− f Ii + τi ) Then:
. z. 1i = z2i z2i = z3i z. = ∆g + g −1 ∆g f + (1 + g −1 ∆g )τ 3i Ii Ii Ii Ii Ii i i
(9)
(10)
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To design the auxiliary control law τi : τi = τInomi + τIsmi
(11)
where τInomi is adopted to achieve finite time stabilization of the nominal part of Formula (8), τInomi is designed as: τInomi = −c I1i sign(z1i )|z1i |1/7 − c I2i sign(z2i )|z2i |1/5 − c I3i sign(z3i )|z3i |1/3
(12)
As long as the choice of c I1i , c I2i , c I3i satisfy the Holwitz condition for polynomial p3 + c I3i p2 + c I2i p + c I1i , the nominal part of Formula (8) can be stabilized in finite time [30]. Sliding mode control item τIsmi is designed to conquer uncertainty. To define sliding mode function σIi : ( σIi = z3i + τauxi (13) . τ auxi = −τInomi Then:
.
1 −1 −1 σ Ii = (1 + g− Ii ∆g ) τIsmi − g Ii ∆g f Ii + ∆ f Ii + gi ∆gτInomi
(14)
τIsmi is designed as: τIsmi = − β Ii sign(σIi )
(15)
Choosing Lyapunov function V (σIi ) = 12 σIi2 to confirm the range of β Ii so as to stabilize the system. h i . . 1 −1 V (σIi ) = σIi σ Ii = σIi (1 + g− ∆g ) τ − g ∆g f + ∆ f − τ Ii i Ii Ii Ii Inomi Ii Ii h i 1 −1 ˆ −1 = σIi −(1 + g− ∆g ) β sign ( σ ) − g g Ii Ii Ii Ii Ii Ii f Ii + ∆ f Ii + g Ii ∆g Ii τInomi
(16)
≤ β Ii |σIi |+(1 − aˆ Ii ) β Ii |σIi |+(1 − aˆ Ii )| f Ii |+∆ f Ii + (1 − aˆ Ii )|τInomi | When β Ii >
(1− aˆ Ii )(|τnomi |+| f Ii |)+∆ f Ii +η Ii aˆ Ii
is satisfied, then:
. V (σIi ) ≤ aˆ Ii β Ii σIi +[(1 − aˆ Ii )(|τnomi |+| f Ii |) + ∆ f Ii ] σIi < η Ii σIi
(17)
τIi = τInomi + τIsmi is substituted to Formula (8) and the equivalent closed loop dynamic similar to nominal integral chain is then procured. Finite time stabilization of system (8) is achieved and third-order sliding mode with respect to s Ii is established. The designed sliding mode control law (14) is based on the assumption that the upper bounds of ∆ f Ii and ∆g Ii are known. However, unmodeled dynamics and the parameter variation of the factual microgrid system are unknown. Uncertain upper bounds are difficult to determine. Control chattering will be increased and power electronic device may be easily damaged if the upper bounds are conservatively set. Therefore, β Ii should be constructed to increase or decrease adaptively, according to the uncertainty variation. The adaptive law of β Ii is designed as: k µ Ii 1 β Ii sign(µ Ii ) Ii . 4 β Ii = ρ β Ii . .. µ Ii = s Ii +τt s Ii +τt2 s Ii −φi τt3
if
β Ii > β Imi
if
β Ii ≤ β Imi
(18)
where φi > 0, τt is the sampling period and ρ β Ii > 0 can be arbitrarily small to guarantee positive of β Ii .
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Theorem 1. Considering system (3), the control laws are designed as Formulas (9), (11), (12), (15), and adaptive control gain is constructed as (18), then the real third-order sliding mode with respect to s Ii will be established in finite time, that is: . .. (19) s Ii ≤ γ0i τt3 , s Ii ≤ γ1i τt2 , s Ii ≤ γ2i τt where γ0i ≥ 0 , γ1i ≥ 0 , γ2i ≥ 0. The microgrid system under islanding operation mode achieves finite time stabilization. Proof. to choose the Lyapunov function: V (σIi ) =
1 2 1 σIi + ( β Ii − β∗Ii )2 2 2
(20)
where β∗Ii is upper bounds of β Ii . The uncertainty of the microgrid system is bounded, thus β∗Ii is bounded. To calculate the first-order time derivative of V (σIi ): .
.
.
= σIi σ Ii + β Ii ( β Ii − β∗Ii ) . = σIi σ Ii + k Ii µ Ii 1 β Ii sign(µ Ii )( β Ii − β∗Ii )
(21)
i h 1 −1 + ∆ f + g τ − f = − β Ii 1 + g− ∆g ∆g ) ( σ Ii σIi Ii Ii nomi Ii Ii h Ii i Ii ≤ − aˆ Ii β Ii σIi + (1 − aˆ Ii )(|τnomi |+| f Ii |) + fˆIi σIi
(22)
V (σIi )
4
.
σIi σ Ii
Consider the following two cases. Case 1: µ Ii > 0. This means that third-order sliding mode with respect to s Ii is not established. (1− aˆ Ii )(|τInomi |+| f Ii |)+ fˆIi i +ηi is satisfied, according to adaptive Control gain β Ii will increase until β Ii > aˆ Ii law (18). Then: . (23) σIi σ Ii ≤ −η Ii1 σIi . V (σIi ) ≤ −η Ii1 σIi −k Ii µ Ii 1 β Iim β Ii − β∗Ii 4 1 2 σ2 | β − β∗ | 2 (24) ≤ −η Ii1 |σIi | − η Ii2 | β Ii − β∗Ii | ≤ −η Ii 2Ii + Ii 2 Ii 1
≤ −η Ii V 2 (σIi ) √ √ where η Ii2 = −k I i µ Ii 1 β Iim , η Ii = min( 2η Ii1 , 2η Ii2 ). Thus, µ Ii < 0 is satisfied in finite 4 time and real third-order sliding mode with respect to s Ii is established in finite time, satisfying s Ii ≤ γ0i τ 3 , s. Ii ≤ γ1i τ 2 , s.. Ii ≤ γ2i τt . t t (1− aˆ Ii )(|τInomi |+| f Ii |)+ fˆIi i +ηi Case 2 µi < 0. Control gain β Ii will decrease and satisfy β Ii ≤ , aˆ .
Ii
according to Formula (18). The signal of V (σIi ) is uncertain and closed-loop stability could not reached. . 1 (1− aˆ Ii )(|τInomi |+| f Ii |)+ fˆIi i +ηi Hence, µi will exceed zero and β Ii > is satisfied. Then V (σIi ) ≤ −η Ii V 2 (σIi ) aˆ Ii is achieved. The establishment of real third-order sliding mode with respect to s Ii means the realization of reference voltage tracking and the tracking error satisfies s Ii ≤ γ0i τt3 . Then the microgrid system under islanding operation mode achieves finite time stabilization. During the control design process, the involved first-order and second-order time derivative of si can be observed via a dynamic gain exact robust differentiator [31].
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3.2. Adaptive Second-Order Sliding Mode Control for Grid-connected Operation of the Microgrid All DGus in the microgrid system work at current control manner under grid-connected operation. The main control objective is to make the VSI output current track the desired value via terminal voltage control and then achieve active and reactive power regulation. Sliding mode variables are chosen as: ( sGd (t) = yGd,re f − yGd (25) sGq (t) = yGd,re f − yGq First-order time derivative is deduced as: .
sGd (t) .
sGq (t)
= hd (t, x ) + gd (t, x )u1 (t) . = yGd,re f + L1t x1 (t) + RLtt x3 (t) − ωx4 (t) − = hq (t, x ) + gq (t, x )u2 (t) . = yGq,re f + L1t x2 (t) + ωx3 (t) +
1 1 Lt u1 ( t ) − Lt uVSId
Rt 1 1 Lt x4 ( t ) − Lt u2 ( t ) − Lt uVSIq
(26)
(27)
The system relative degrees are one. ITd and ITq can track the reference values by designing a first-order sliding mode control law for Formulas (26) and (27). However, the control chattering is severe, which is harmful to VSI operation and has a strong impact on the lifetime of a power electronic device. Thus, the system relative degree is enhanced to two, meanwhile, system modeling error and parameter disturbance are considered. Then: (
..
sGi (t) = f i ( x (t), u(t), di (t)) + gi vi (t) . ui ( t ) = vi ( t ) i = d, q
(28)
where vi (t) is the first-order time derivative of actual control variable ui (t). f i and gi are respectively represented as: f d ( x (t), u(t), dd (t)) = −
+
1 RLt C
+
2ωRt 1 L t x4 ( t ) − L t C x5 ( t ) −
Rt x1 ( t ) + L2t 1 L t C x7 ( t ) +
R2t 2ω 1 2 L t x 2 ( t ) + ω + L t C − L2 x 3 ( t ) t .. Rt ω u ( t ) + x (t) + dd (t) u ( t ) − q 3,re f d 2 Lt L
(29)
t
= f d + dd (t) f q ( x (t), u(t), dq (t)) = − 2ω L t x1 ( t ) − 2 R + ω 2 + L1t C − L2t x4 (t) −
+
..
t
1 RLt C
+
Rt L2t
x2 ( t ) −
2ωRt L t x3 ( t )
1 1 ω L t C x6 ( t ) − L t C x8 ( t ) + L t u d ( t )
Rt u (t) + x4,re f (t) + dq (t) L2t q
(30)
= f q + dq (t) gd = gq = −
1 Lt
(31)
dd (t) =
Rt ω 1 . uSV Id − uSV Iq + uSV Id + ∆d Lt Lt L2t
(32)
dq (t) =
ω 1 . Rt uSV Iq + uSV Id + uSV Iq + ∆q Lt Lt L2t
(33)
where ∆d and ∆q are the parameter uncertainties. The nominal value and signal of gi is known in advance. Even so, there is an uncertain part for Lt . Thus, Formula (28) can be represented as: ..
sGi (t)
= f i ( x (t), u(t), di (t)) + gi vi (t) = f i ( x (t), u(t)) + di (t) + ( gi + gˆi )vi (t)
(34)
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di (t) and gˆi (i = d,q) are both bounded and the upper bounds are unknown in the microgrid −1 system and satisfy g gˆi ≤ 1 − ai , di (t) ≤ di , 0 < ai ≤ 1. Second-order sliding mode control with i
respect to sGi is equivalently viewed as a finite time stabilization problem: (
.
z1i = z2i . z2i = f i ( x (t), u(t)) + di (t) + ( gi + gˆi )vi (t)
(35)
.
where z1i = sGi and z2i = sGi . Considering state feedback control: vi = gi −1 (− f i ( x (t), u(t)) + τi ) Then:
(
(36)
.
z1i = z2i . z2i = di (t) + gi −1 gˆi f i ( x (t), u(t)) + (1 + gi −1 gˆi )τi (t)
(37)
To design: τGi = τGnomi + τGsmi
(38)
where τGnomi is adopted to achieve finite time stabilization of the nominal part of Formula (37), τGnomi is designed as: τGnomi = −cG1i sign(z1i )|zi1 |1/3 − cG2i sign(z2i )|z2i |1/5 (39) As long as the choices of cG1i , cG2i satisfy the Holwitz condition for polynomial p2 + cG2i p + cG1i , the nominal part of Formula (37) can be stabilized in finite time [30]. Sliding mode control item τGsmi is designed to conquer uncertainty. To define sliding mode function σGi : ( σGi = z2i + τauxi (40) . τ auxi = −τGnomi .
ˆ Gnomi σ Gi = (1 + gi−1 gˆ )τGsmi − gi−1 gˆ f i + di (t) + gi−1 gτ
(41)
τsmi = − β Gi sign(σGi )
(42)
To design: 2 to confirm β Choosing the Lyapunov function V (σGi ) = 12 σGi Gi so as to stabilize the system.
h i . . V (σGi ) = σGi σ Gi = σGi (1 + gi−1 gˆi )τGi − gi−1 gˆi f + d(t) − τGnomi h i = σGi −(1 + gi−1 gˆi ) β i sign(σGi ) − gi−1 gˆi f + d(t) + gi−1 gˆi τGnomi ≤ β i σGi +(1 − ai ) β i σGi +(1 − ai ) f i +di + (1 − ai ) τGnomi When β Gi >
(1− ai )(|τnomi |+| f i |)+di +ηi ai
(43)
is satisfied, then:
h i . V (σGi ) ≤ ai β i σGi + (1 − ai )( τGnomi + f i ) + di σGi < ηi σGi
(44)
τGi = τGnomi + τGsmi is substituted to Formula (37) and an equivalent closed loop dynamic similar to nominal integral chain is then procured. Finite time stabilization of system (37) is achieved and second-order sliding mode with respect to sGi is established. The designed sliding mode control law (42) is based on the assumption that the upper bounds of ai and di are known. However, unmodeled dynamics and parameter variation of factual the microgrid system are unknown. Uncertain upper bounds are hard to been determined. Control chattering will be increased and the power electronic device may be easily damaged if the upper bounds are
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conservatively set. Therefore, β Gi should be constructed to increase or decrease adaptively according to the uncertainty variation. Adaptive law of β Gi is designed as: .
βGi
k Gi µGi 41 β i sign(µGi ) = ρ βi . µGi = si +τt si −φi τt2
if
β i > β mi
if
β i ≤ β mi
(45)
Theorem 2. Considering system (4), the control laws are designed as Formulas (36), (38), (39), (42), and adaptive control gain is constructed as (45), then the real second-order sliding mode with respect to sGi will be established in finite time, that is: . sGi ≤ γ0Gi τt2 , sGi ≤ γ1Gi τt
(46)
where γ0Gi ≥ 0, γ1Gi ≥ 0. The microgrid system under grid-connected operation mode achieves finite time stabilization. Proof. to choose Lyapunov function: 1 1 2 σ + ( β − β∗Gi )2 2 Gi 2 Gi
V (σGi ) =
(47)
where β∗Gi is upper bounds of β Gi . Microgrid uncertainty under grid-connected mode is bounded, thus β∗Gi is bounded. To calculate first-order time derivative of V (σGi ): .
.
.
= σGi σ Gi + βGi ( β Gi − β∗Gi ) . = σGi σ Gi + k Gi µGi 1 β Gi sign(µGi )( β Gi − β∗Gi )
(48)
i h = − β Gi 1 + gi−1 gˆi σGi + gi−1 gˆi τnomi − f i + di σGi h i ≤ − ai β Gi σGi + (1 − ai ) τnomi + f i + di σGi
(49)
V (σGi )
4
.
σGi σ Gi
Considering the following two cases. Case 1: µGi > 0. This means that second-order sliding mode with respect to sGi is not established. (1− ai )(|τnomi |+| f i |)+di +ηi Control gain β Gi will increase until β Gi > is satisfied according to adaptive ai law (45). Then: . σGi σ Gi ≤ −ηGi1 σGi (50) . V (σGi ) ≤ −ηGi1 σGi −k Gi µGi 1 β Gim β Gi − β∗Gi 4 1 2 2 2 β Gi − β∗Gi | σ | ∗ Gi (51) ≤ −ηGi1 σGi −ηGi2 β Gi − β Gi ≤ −ηGi 2 + 2 1
≤ −ηGi V 2 (σGi ) √ √ where ηGi2 = −k Gi µGi 1 β Gim , ηGi = min( 2ηGi1 , 2ηGi2 ). Thus, µGi < 0 is satisfied in finite 4 time and real second-order sliding mode with respect to sGi is established in finite time satisfying sGi ≤ γ0Gi τ 2 , s. Gi ≤ γ1Gi τ . t t (1− ai )(|τnomi |+| f i |)+di +ηi Case 2 µGi < 0. Control gain β Gi will decrease and satisfy β Gi ≤ according a .
i
to Formula (45). The signal of V (σGi ) is uncertain and closed-loop stability could not be reached.
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β Gi ≤
(1 − ai )(| τ nomi | + | fi |) + d i + ηi ai
11 of 17 reached. Hence, μi will exceed zero and is satisfied. Then 1 V (σ Gi ) ≤ −ηGiV 2 (σ Gi ) is achieved. . 1 (1− ai )(|τnomi |+| f i |)+di +ηi Hence, µ will exceed zero and is satisfied. Then differentiator V (σGi ) ≤ −ηGi[31]. V 2 (σThe i Gi ≤ Gi ) The involved sGi can beβ observed via a adynamic gain exact robust i is achieved. establishment of real . second-order sliding mode with respect to sGi means the realization of The involved sGi can be observed via a dynamic gain exact robust differentiator [31]. | sGi |≤ γ 0Giτtot2 .s Then reference voltage tracking and the tracking errormode satisfies the microgrid system The establishment of real second-order sliding with respect means the realization of Gi 2 under grid-connected operation mode achieves time sstabilization. □ ≤ γ τ reference voltage tracking and the tracking errorfinite satisfies . Then the microgrid system Gi 0Gi t under grid-connected operation mode achieves finite time stabilization. | g Ii − 1 Δ g Ii |≤ 1 − aˆ Ii During the design of HOSM controllers, as soon as the uncertain terms satisfy −1 ˆ the uncertain terms satisfy g Ii ∆g Ii ≤ 1 − aˆ Ii , aˆ Ii ≤ 1 ,of| HOSM 0 < a ≤ 1 0
