electronics

Article Control Design of LCL Type Grid-Connected Inverter Based on State Linearization

Longyue Yang 1,2,*, Chunchun Feng 2 and Jianhua Liu 1,2

1 Jiangsu Province Laboratory of Mining Electric and Automation, China University of Mining and Technology, Xuzhou 221116, China 2 School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou 221116, China * Correspondence: [email protected]; Tel.: +86-0516-83885605

 Received: 11 July 2019; Accepted: 3 August 2019; Published: 7 August 2019 

Abstract: Control strategy is the key technology of power electronic converter equipment. In order to solve the problem of controller design, a general design method is presented in this paper, which is more convenient to use computer machine learning and provides design rules for high-order power electronic system. With the higher order system Lie derivative, the nonlinear system is mapped to a controllable standard type, and then classical linear system control method is adopted to design the controller. The simulation and experimental results show that the two controllers have good steady-state control performance and dynamic response performance.

Keywords: state feedback linearization; grid-connected inverter; LCL filter

1. Introduction As the main equipment of an AC power system, grid-connected voltage source inverters (VSI) have been widely used in recent years, such as photovoltaic inverters [1], Pulse Width Modulation Pulse Width Modulation (PWM) rectifiers [2], static var generators [3], active power filters [4], etc. The filter is an important part of the inverter, the structure of which directly determines the mathematical model and control mode of the inverter. Nowadays, a Inductance-Capacitance-Inductance (LCL) filter, with the advantages of small size, low cost and high harmonic attenuation for high frequency current, is widely used in voltage source type grid-connected converters. Considering that the LCL filter is the 3-order system, the damping of the system is small, resulting in a resonant peak of the grid-side inductance current. This phenomenon will adversely affect the safe and stable operation of grid-connected system. In order to improve the damping characteristics of the system, additional system damping is required. Active damping control method is commonly used at present stage [5]. With the state variables feedback, the traditional proportional integral controller can be used to achieve the grid-side inductance current control. However, for nonlinear power electronic devices, this design method of the controller is based on small signal modeling [6] and harmonic linearization. As the coupling and high order terms in the system are ignored in the Taylor series calculation, the obtained controller is suitable for working at steady working point with poor performance in other control domains [7]. In order to solve the global control problem, many solutions have been put forward [8,9]. Among them, the state feedback linearization method that develops from differential geometry [10,11] has become an effective way to solve the nonlinear power electronics system control problems. Based on the differential homeomorphism, Lie derivative is used to analyze the numerical relation between the state variables, the input variables, and the output variables. Then, the necessary and sufficient conditions for controllability and observability of nonlinear control systems can be established. Choi et

Electronics 2019, 8, 877; doi:10.3390/electronics8080877 www.mdpi.com/journal/electronics 2 of 16 theElectronics permanent2019, 8magnet, 877 synchronous motor [12]. In addition, the drive flux and torque ripple2 of were 15 suppressed. Yang et al. combined feedback linearization and sliding mode variable structure controlal. proposed to complete a feedback the control linearization of three-phase direct torque four-leg control inverter for the permanent [13]. This magnet method synchronous was used to decouplemotor [the12]. torque In addition, and stator the drive flux fluxof the and inductiv torquee ripple motor were by Lascu suppressed. et al. [14]. Yang Yang et al. et combined al. applied thefeedback state feedback linearization control and to sliding the modeModular variable Multil structureevel Converter control to complete(MMC) thesystem control and of three-phaseanalyzed the performancefour-leg inverter characteristics [13]. This of method the system was used [15]. to decouple the torque and stator flux of the inductive motorDifferent by Lascu from et al.the [14 local]. Yang approximate et al. applied lineariz the stateation feedback method, control the to thefeedback Modular decoupling Multilevel is achievedConverter by (MMC)adopting system this nonlin and analyzedear algorithm the performance without ignoring characteristics the higher-order of the system terms. [15]. However, the commonlyDifferent used from thefeedback local approximate linearization linearization algorithm method, at the thepresent feedback stage decoupling is mostly is used achieved in the by 2- orderadopting or 1-order this nonlinear system. algorithm The rese withoutarch on ignoring the controller the higher-order design of terms. high-order However, or the other commonly complex systemsused feedbackis rare. linearizationIn addition, algorithmwhen the at order the present of the stage controlled is mostly object used is in high, the 2-order whether or 1-orderthe state feedbacksystem. control The research can be on transformed the controller into design a simple of high-order form has or other not been complex analyzed systems by is literatures. rare. In addition, whenIn order the order to achieve of the controlled the control object of high-order is high, whether powe ther electronic state feedback systems, control the can design be transformed of controller into a simple form has not been analyzed by literatures. based on LCL filter type grid-connected inverters is studied in this paper. For the 3-order control In order to achieve the control of high-order power electronic systems, the design of controller system, two controller design methods based on state feedback linearization are proposed in this based on LCL filter type grid-connected inverters is studied in this paper. For the 3-order control paper. Lie derivative vector field is used to solve the system relationship and design the decoupling system, two controller design methods based on state feedback linearization are proposed in this paper. matrix. Through the nonlinear mapping, the nonlinear system can be mapped to a controllable Lie derivative vector field is used to solve the system relationship and design the decoupling matrix. standardThrough form. the nonlinearThen, the mapping,classical linear the nonlinear system systemcontrol can method, be mapped which to can a controllable be designed standard easily, is applicableform. Then, for the classicalcontroller linear design. system Finally, control the method, performance which canof the be designedtwo controllers easily, is is applicable compared for and analyzedthe controller by simulations design. Finally, and experiments. the performance of the two controllers is compared and analyzed by simulations and experiments. 2. Single Closed-Loop Controller Design Based on State Feedback Linearization 2. Single Closed-Loop Controller Design Based on State Feedback Linearization 2.1. Model of the Three-Phase Three-Leg Grid-Connected Inverter 2.1. Model of the Three-Phase Three-Leg Grid-Connected Inverter Figure 1 shows the structure of the three-phase three-leg LCL grid-connected inverter. L1 and Figure1 shows the structure of the three-phase three-leg LCL grid-connected inverter. L1 and L2 L2 denote the inverter side inductance and the grid-side inductance, respectively. denote the inverter side inductance and the grid-side inductance, respectively.

L L i 1 2 a e i a + b e U b dc − ic ec C

Figure 1. Schematic diagram of the three-phase three-leg grid-connected inverter. Figure 1. Schematic diagram of the three-phase three-leg grid-connected inverter. As the model has been analyzed by many literatures, this paper does not repeat the description. TheAs mathematical the model modelhas been of inverter analyzed in dq coordinateby many axisliteratures, is given directlythis paper as does not repeat the description. The mathematical model of inverter in dq coordinate axis is given directly as    i  0 ω 1/L 0 0 0 m /L  i   1d   ω − 1 d 1  1d     01000− LmL11d    i  ii ω 0 0 1/L1 0 0 mq/ L1  i   1q  11dd −−ω  1q     − 00100− LmL11q    ucd  ii11qq 1/C 0 0 ω 1/C 0 0  ucd     ω −−   d  u  = 0100CC 1/C ω 0 100 0 1/C 0  u  dt  cq  uucd  cd  cq     01CC− −−ω 0 01− 0    i d  = 0 0 1/L2 0 0 ω 0  i   2d  uucq   cq  2d  (1)  dt  00100L ω 0    i2q   0 0 02 1/L2 ω 0 0  i2q    ii22dd      md mq −ω−   (1) u  00010 0L2 0 00 0 0 u dc iiCdc Cdc dc 22qq− − T  " md mq # " # 0−− 0 0 0 0000001/L2 0 e  uudc+ d  dc CCdc dc − 0 0 0 0 1/L 0 0 eq − 2 0000 0− 1L 0T e + 2 d −  0000 1L2 0 0 eq

Electronics 2019, 8, 877 3 of 15

h iT where the state variables X = i1d i1q ucd ucq i2d i2q udc represent the inverter side inductance current, the filter capacitor voltage, the grid-side inductance current and the DC voltage in the synchronous reference coordinate system, respectively. ed and eq represent grid voltage in the synchronous reference coordinate system, respectively. md and mq represent the nonlinear pulse width modulation variables, respectively. ω represents the fundamental angular frequency of the system. According to (1), due to the mutual inductance and mutual capacitance, the inductance current and the capacitance voltage is coupled in the synchronous reference coordinate system. Usually, the voltage source grid-connected inverter system is a double closed loop structure. The outer loop controls DC voltage, and the inner loop controls AC current. However, for the different structure like the PV system, the DC voltage may not need to be controlled. Considering the generality, only the current inner loop is analyzed. Due to the LCL filter, the system can be regarded as a 2-input 2-output system. At the same time, the existence of LCL filter capacitor, which introduces a pair of conjugate pure imaginary roots to cause resonance, increases the order of the system. Therefore, it is necessary to consider the suppression of resonance.

2.2. Single Closed-Loop Control Strategy Based on State Feedback Linearization Define the LCL filter inverter side inductance current, filter capacitor voltage, and grid-side T h iT inductance current as state variables, written as X = [x1 x2 x3 x4 x5 x6] = i1d i1q ucd ucq i2d i2q . Define the 2 dimensional modulation vector in the synchronous coordinate system as the input variable, T h iT written as U = [u1 u2] = md mq . The deviation of grid-side inductance current (controlled object) T h iT is taken as the output variable Y = R(X) = [r (X) r (X)] = i i i i q , in which i and 1 2 dre f − 2d qre f − 2 dre f iqre f are the reference current. Therefore, Equation (1) can be expressed as a nonlinear differential equation composed of polynomials of state variables and input variables:  .  X = f[X(t)] + G[X(t)]U  (2)  Y = R(X)

   ωx2 x3/L1   −   ωx x /L   1 4 1   − −  " #T  x1/C + ωx4 x5/C  0 udc/L1 0 0 0 0 where f(X) =  − , G(X) = [g1(X)g2(X)] = .  x /C ωx x /C  u /L 0 0 0 0 0  2 3 6  dc 1  − −   x3/L2 + ωx6 ugd/L2   −  x /L ωx ugq/L 4 2 − 5 − 2 Equation (2) shows that the LCL-type three-phase three-leg converter is a 2-input 2-output affine nonlinear system with the state variables number of n = 6. It is nonlinear for the state vector X, but linear for the input U. Since the state variables are all dq-axis symmetric variables, the number of vector field pairs is l = 3. The corresponding l 1 order Lie derivative can be described as −  ad g (X) = L g L f  f 1 f 1 g1  −  h ωudc udc iT  = 0 0 0 0  L1 − L1C  2  ad g1(X) = L f [ad f g1(X)] Lad g (X) f  f − f 1   2 T  ω udc udc 2ωudc udc  = + 2 0 0 0  L1 L C L1C − L1L2C  1 (3)  ad f g2(X) = L f g2 Lg f  − 2  h iT  ωudc udc  = L 0 0 L C 0 0  − 1 − 1  ad2 g (X) = L [ad g (X)] L f  f 2 f f 2 ad f g2(X)  −   2 T  = 0 ω udc + udc 2ωudc 0 0 udc  L1 L2C L1C L1L2C 1 − − Electronics 2019, 8, 877 4 of 15

where ad g denotes Lie bracket operation of vector fields f and g, written as ad g = L g Lg f = f f f − (∂g/∂X) f (∂ f /∂X)g. Then, the corresponding 6-dimensional vector field matrix can be expressed as − h 2 2 i D6 = g1 g2 ad f g1 ad f g2 ad f g1 ad f g2  2   udc 0 0 ωudc ω udc + udc 0   L1 L1 L1 L2C   − 1   2   udc ωudc ω udc udc   0 0 0 +   L1 L1 L1 L2C   1   udc 2ωudc  (4) =  0 0 0 0   − L1C − L1C   udc 2ωudc   0 0 0 0   − L1C L1C   udc   0 0 0 0 L L C 0   − 1 2 u   0 0 0 0 0 dc  − L1L2C

Obviously, it can be obtained that RankD6 = 6 = n. The rank of the matrix formed by the vector fields is n in the neighborhood of X . 0   2 2 According to Equations (3) and (4), the elements in g1 g2 ad f g1 ad f g2 ad f g1 ad f g2 do not contain state variables. In other words, they are the constant vector fields, meaning that the result of Lie brackets operation between two elements is a zero vector. Define Di as the matrix of the former i column elements in Equation (4). It can be calculated that the 6 vector fields satisfy involution relation. Therefore, according to the nonlinear , state feedback can be applied to linearize the inverter system. A MIMO nonlinear system has a relative degree {r1, ... , rm} at a point x0 if

k 1. Lg L r (X) , 0 for all 0 j m, for all k r 1, for all 0 i m and for all x in a neighborhood j f i ≤ ≤ ≤ i − ≤ ≤ of x0. 2. the m m matrix E(X) is nonsingular at x = x . × 0 In Equation (2), the output variable of the inverter is current error r (X) = i i . In order to 1 dre f − 2d analyze the numerical relation of the output, the Lie derivatives need to be calculated as follows:

 ∂r (X)  L r (X) = 1 g (X) = 0  g1 1 ∂X 1   ∂r1(X)  Lg r1(X) = g2(X) = 0  2 ∂X  ∂r (X)  L r (X) = 1 f (X) = 1 x ωx + 1 e  f 1 ∂X L2 3 6 L2 d  − −  ∂(L f r1(X))  L L r (X) = g (X) = 0  g1 f 1 ∂X 1   ∂(L f r1(X)) (5)  Lg2 L f r1(X) = g2(X) = 0  ∂X  ∂(L r (X))  2 f 1 x5 x1 2ω 2 ω  L r1(X) = f (X) = − x4 + ω x5 + eq  f ∂X L2C − L2 L2  2  ∂(L r1(X))  2 f udc  Lg L r1(X) = g1(X) = , 0  1 f ∂X − L1L2C  2  ∂(L r1(X))  2 ( ) = f ( ) =  Lg2 L f r1 X ∂X g2 X 0

2 ( ) = where Lg1 L f r1 X , 0. The relation degree of the output is d1 3. Similarly, the output r (X) = i i q is symmetrical. The relation degree of the output r (X) 2 qre f − 2 2 is d2 = 3. Electronics 2019, 8, 877 5 of 15

The Lie derivatives can be obtained as follows:

 ∂r2(X)  Lg r2(X) = g1(X) = 0  1 ∂X  r (X)  L r (X) = ∂ 2 g (X) =  g2 2 ∂X 2 0   ∂r2(X) 1 1  L r (X) = f (X) = x + ωx + eq  f 2 ∂X L2 4 5 L2  −  ∂(L f r2(X))  L L r (X) = g (X) = 0  g1 f 2 ∂X 1   ∂(L f r2(X)) (6)  Lg2 L f r2(X) = g2(X) = 0  ∂X  ∂(L r (X))  2 ( ) = f 2 ( ) = x6 x2 + 2ω + 2 ω  L r2 X X f X L−C L x3 ω x6 L ed  f ∂ 2 2 − 2  ∂(L2 r (X))  2 f 2  Lg L r2(X) = g1(X) = 0  1 f ∂X  2  ∂(L r2(X))  2 f udc  Lg L r2(X) = g2(X) = , 0 2 f ∂X − L1L2C Based on the above analysis, the decoupling matrix can be established as follows.

 2 2   Lg L r1(X) Lg L r1(X)   1 f 2 f  E(X) =  2 2   Lg1 L r2(X) Lg2 L r2(X)  " f f # (7) u /(L L C) 0 = − dc 1 2 0 u /(L L C) − dc 1 2 It can be calculated that the matrix is a non-singular matrix. The total relation degree of the system output variables is d = d1 + d2 = 6 = n. Therefore, in view of coordinate transformation, the original system can be directly transformed into a controllable linear system of Brunovsky standard type. Define the state variable of linear standard system after feedback linearization as T Z = [z1 z2 z3 z4 z5 z6] . According to the relative order theory, the relationship between the new state variables and the original state variables can be expressed as

 = ( ) =  z1 r1 X idre f x5  −  1 1  z2 = L f r1(X) = L x3 ωx6 + L ed  − 2 − 2  2 x5 x1 2ω 2 ω  = (X) = − + +  z3 L f r1 L C L x4 ω x5 L eq  2 − 2 2 (8)   z4 = r2(X) = iqre f x6  −  1 1  z = L r (X) = x + ωx + eq  5 f 2 L2 4 5 L2  −  2 x6 x2 2ω 2 ω  z6 = L r2(X) = − + x3 + ω x6 ed f L2C L2 − L2 Equation (8) shows that the original system state variable X is mapped into new state variable Z after the coordinate transformation of the Lie derivative. The coupling and high order terms in the system are not ignored in the coordinate transformation. The numerical relationship between new state variables can be calculated as Electronics 2019, 8, 877 6 of 15

.  . ∂r1(X) 1 1  z = X = x3 ωx6 + e = z2  1 ∂X L2 L2 d  − −  . ∂(L r (X)) .  f 1 x5 x1 2ω 2 ω  z2 = X = − x4 + ω x5 + eq = z3  ∂X L2C − L2 L2  2  . ∂(L r1(X)) . 2  f 3ω L1+L2 3ω  z3 = X = x2 + ( + )x3  ∂X L2C L L2C L2  − 1 2  3ω 3 1 ω2 udc  +( + ω )x6 ( + )e u  L2C L2C L2 d L1L2C 1  − 2 − . . (9)  ∂r2(X) 1 1  z = X = x + ωx5 + eq = z5  4 ∂X L2 4 L2  −  . ∂(L r (X)) .  f 2 x6 x2 2ω 2 ω  z5 = X = − + x3 + ω x6 ed = z6  ∂X L2C L2 − L2  2  . ∂(L r2(X)) . 2  f 3ω L1+L2 3ω  z6 = X = x1 + ( + )x4  ∂X L2C L L2C L2  1 2  3ω 3 1 ω2 udc  ( + ω )x ( + )eq u  L2C 5 2 L2 L1L2C 2 − − L2C −

It can be seen that information of z1 is not directly contained in state variable z3 and information of z4 is not directly contained in state variable z6. Defining the control variable after the coordinate T transformation as V = [v1 v2] , the relationship between the original nonlinear system control variable U and V of can be obtained as 1 U = E− (X)[V Γ(X)] (10) − 2   3ω L1+L2 3ω    x2 + ( + )x3    L2C L L2C L2     − 1 2       2   r1 ( )   +( 3ω + ω3)x ( 1 + ω )e    L r1 X   L2C 6 L2C L2 d   f   − 2  where Γ(X) =  r  =   2  .  2   3ω L1+L2 3ω   L r2(X)    x1 + ( + )x4   f   L2C L L2C L2     1 2     3ω 3 1 ω2     ( + ω )x ( + )eq   L2C 5 2 L2 − − L2C Substituting Equation (10) into Equation (2), the controller can be expressed as

 L +L u = 1 [ 3L ωi + (3L Cω2 + 1 2 )u  1 u 1 1q 1 L2 cd  dc −  3 2 L1 L1L2C  +(3L1ω + L1L2Cω )i2q (L1Cω + )ed] v1  − L2 − udc (11)  1 2 L1+L2  u2 = [3L1ωi1d + (3L1Cω + L )ucq  udc 2  3 2 L1 L1L2C  (3L1ω + L1L2Cω )i2d (L1Cω + )eq] v2 − − L2 − udc The original nonlinear system is transformed into the linear system as follows:  .  Z = AZ + BV  (12)  Y = CZ

      0 1 0 0 0 0 0 0 1 0              0 0 1 0 0 0   0 0   0 0         0 0 0 0 0 0   1 0   0 0  A =   B =   C =   where  ,  ,  .  0 0 0 0 1 0   0 0   0 1         0 0 0 0 0 1   0 0   0 0         0 0 0 0 0 0   0 1   0 0  The matrix A shows that the dq components of the original system are completely decoupled by nonlinear mapping. The of the system is shown in Figure2. The block diagrams in the dotted box are the equivalent controller and the original system. 7 of 16

 1 LL+ uLiLCu=−[3ωω + (32 +12 )  1111uLqcd  dc 2  ++ωω32 − ω +LLLC112 −  (3L1122LLC ) iqd ( LC 1 ) e ] v 1  Lu2 dc  + (11)  =++1 ωω2 LL12 uLiLCu2111[3dcq (3 )  uLdc 2  LLLC  −+(3L ωωLLC32 ) i − ( LC ω +112 ) e ] − v  1122dq 1 2  Lu2 dc

The original nonlinear system is transformed into the linear system as follows:

ZAZBV =+  (12) YCZ=

010000 00 10   001000 00 00 000000 10 00 where A = , B = , C = . 000010 00 01 000001 00 00   000000 01 00 The matrix A shows that the dq components of the original system are completely decoupled Electronicsby nonlinear2019, 8mapping., 877 The transfer function of the system is shown in Figure 2. The block diagrams7 of 15 in the dotted box are the equivalent controller and the original system.

ed i1q i2q −−ω2 − ω ωω+ 3 LC112 L L 3L1 3LLLC112 * i u i i u i 2d − 1d 1d cd cd 2d v1d LLC12 1 udc udc 1/sL1 1/ sC 1/ sL2 i 2d u ω2 ++ cd 3()LC1122 L L L

eq i1d i2d −−ω2 − ω ωω+ 3 LC112 L L 3L1 3LLLC112 * i2q u i i ucq i2q v − 1q 1q cq 1q LLC12 1 udc udc 1/sL1 1/ sC 1/sL2 i 2q u ω2 ++ cq 3()LC1122 L L L

FigureFigure 2. 2. CurrentCurrent control control system system based based on on state state feedback feedback linearization. linearization.

For the system system shown shown in in Equation Equation (12), (12), the the controller controller V canV canbe designed be designed to make to makeit stable it stableat the at T T thepoint point [𝑧 𝑧[z]1 z=4] [0= 0][0. As 0] .the As feedback the feedback decoupling decoupling of the of original the original system system has been has completed, been completed, the thecontroller controller can canbe designed be designed based based on the on linear the linear theory theory to achieve to achieve the control the control of original of original system. system.

2.3.2.3. Parameters Design Design of of Current Current Single Single Closed Closed Loop Loop Controller Controller Through the state feedback feedback linearization, linearization, the the system system is istransformed transformed into into a 3-order a 3-order controllable controllable standardstandard type.type. Therefore,Therefore, at at least least two two open-loop open-loop zeros zeros should should be introduced be introduced into the into linear the controller,linear thatcontroller, is, two first-orderthat is, two di fffirst-orerentiationder differentiation elements. Considering elements. that Cons theidering differentiation that the elements differentiation may cause systemelements noise, may the cause controller system should noise, containthe controller a filter shou partld (i.e., contain the first-order a filter part inertial (i.e., element). the first-order To make theinertial controller element). simple, To make the transfer the controller function simple, of the systemthe transfer controller function after of linearization the system controller can be written after as: linearization can be written as: v = v = (k s2 + k s + k )/(s + k ) (13) 1 2 − 2 1 0 3 Then the open-loop transfer function and the closed-loop transfer function of the control system can be represented as follows: k s2 + k s + k G (s) = 2 1 0 (14) OL 3 s (s + k3) k s2 + k s + k G (s) = 2 1 0 (15) CL 4 3 2 s + k3s + k2s + k1s + k0 Compared with the original open-loop system, the controller which adds two zeros and one pole as a series correction link turns the original 3-order system into a 4-order one. Therefore, the order reduction processing of the higher order system is considered. Assume two poles and two zeros constitute a pair of dipoles. Then, Equation (15) can be rewritten as 2 2 k2(s + 2ζ1ωn1s + ω ) G (s) = n1 (16) CL ( 2 + + 2 )( 2 + + 2 ) s 2ζ2ωn2s ωn2 s 2ζ3ωn3s ωn3 With dipoles elimination, the system can be transformed to a typical 2-order system. For the closed-loop transfer function of the 2-order linear system, the parameter design method of the linear system can be used to construct the restrictive conditions, so as to obtain the parameters range. Electronics 2019, 8, 877 8 of 15

1. Stability conditions of the system According to the stability criterion, the closed loop characteristic equation of the original system should satisfy the Hurtwitz criterion as follows:   ∆1 = k2k3 > 0   ∆ = ∆ k > 0  2 1 − 1 (17)  = k k k2  ∆3 1∆2 0 3 > 0  −  ∆4 = k0k1k2k3 > 0

2. Restrictive conditions of dipoles parameters The closed loop system (16) can be reduced to a 2-order system with sufficient condition of dipoles existence. It is necessary to keep the dipole far from the imaginary axis (i.e., ζ2ωn2 > ζ3ωn3).

q q 2 2 ζ2ωn2 (ζ1ωn1 ζ2ωn2) + j(ωn1 1 ζ ωn2 1 ζ ) < (18) − − 1 − − 2 10

3. Cut-off frequency of the closed loop system

The cut-off frequency (i.e., ωb) of current control system is generally less than 1/5 of switching frequency.

Gain(jω) ω=ωb k = q 2 = 0.707 (19) 2 (ω2 ω2) +(2ζ ω ω )2 n3− b 3 n3 b 4. Closed loop amplitude frequency characteristic at zero frequency The closed loop system should maintain good tracking performance at low frequency band.

The amplitude frequency characteristics satisfy the requirement of Gain(jω) 1. ω=0 ≈ 2 k2ωn1 Gain(jω) = = 1 (20) |ω 0 2 2 ≈ ωn2ωn3

Combined with the above four conditions, the optimal control shown in Equation (12) can be realized according to the classical control theory.

3. Design of Double Closed Loop Controller Based on Reduced Order State Feedback Linearization The single closed loop feedback linearization method is used to decouple and simplify the inverter. The controller designed through this method is simple in structure, but it is too dependent on system precise model. Furthermore, the design of controller coefficient is complex, which limits the application of the algorithm. Considering the objective of control, it is not necessary to configure all open loop poles of LCL type inverter model to the coordinate origin. Therefore, the system can be divided into two parts, which are analyzed separately. Then, a control strategy based on reduced order state feedback linearization can be designed. Consequently, the capacitance voltage in the LCL filter can be adopted as the intermediate variable of the reduced order feedback line linearization system. At the same time with system decoupling control, the single state variable active damping strategy is added to the system to achieve resonance suppression.

3.1. State Feedback of Inverter Side Inductance and Filter Capacitor Subsystem Define the inverter side inductance current and the filter capacitor voltage as state variables T h iT h iT X = [x1 x2 x3 x4] = i1d i1q ucd ucq , modulation variables as input variables U = u1d u1q , filter h iT capacitor voltage as output variables Y = R(X) = ucd ucq . Electronics 2019, 8, 877 9 of 15

The model of the inverter inductance and the filter capacitance subsystem can be written as      ωx2 x3/L1   udc/L1 0   −     ωx x /L  h i  0 u /L  f(X) =  1 4 1  G(X) = g (X) g (X) =  dc 1  Equation (2), where  − − , 1 2  .  x1/C + ωx4 i2d/C   0 0   −    x /C ωx i q/C 0 0 2 − 3 − 2 The grid-side inductance current can be considered as a measurable feedback variable in the system. The model is a 2-input 2-output affine nonlinear system with 4 state variables (i.e., n = 4). According to the aforementioned analysis method, the 4-dimensional vector field matrix can be calculated as follows: h i D4 = g1 g2 ad f g1 ad f g2  u /L 0 0 ωu /L   dc 1 dc 1   −   0 udc/L1 ωudc/L1 0  (21) =    0 0 u /L C 0   − dc 1   0 0 0 u /L C  − dc 1

From (21), it can be obtained that RankD4 = 4 = n. The calculation result shows that the vector fields are involution, which satisfies the state feedback linearization condition. The Lie derivative of the system output (i.e., r1(X) = ucd) can be calculated as  ∂r1(X)  Lg r (X) = g (X) = 0  1 1 ∂X 1  ∂r1(X)  Lg r1(X) = g2(X) = 0  2 ∂X  ( ) = ∂r1(X) ( ) = 1 + 1  L f r1 X ∂X f X C x1 ωx4 C i2d (22)  ( ( )) −  ∂ L f r1 X udc  Lg L f r1(X) = g1(X) = , 0  1 ∂X L1C  ∂(L r (X))  ( ) = f 1 ( ) = Lg2 L f r1 X ∂X g2 X 0

In Equation (22), the total relationship of the output variables of the system is d = d1 + d2 = 4, so the subsystem can be directly transformed to a linear controllable system through coordinate transformation. The corresponding decoupling matrix can be written as

" #  u  L L r (X) L L r (X) dc 0 g1 f 1 g2 f 1  L1C  E(X) = =  udc  (23) Lg L r (X) Lg L r (X)  0  1 f 2 2 f 2 L1C

Then, the new state variables after the feedback linearization can be expressed as

 . r (X) .  z = ∂ 1 X = 1 x + x 1 i  1 ∂X C 1 ω 4 C 2d  ∂(L r (X)) . − 2  . f 1 2ω 1+L1Cω ω udc  z = X = x x i q + u  2 ∂X C 2 L1C 3 C 2 L1C 1d  . ∂r (X) . − − (24)  z = 2 X = 1 x ωx 1 i  3 ∂X C 2 3 C 2q  . ∂(L r (X)) . − − 2  f 2 2ω 1+L1Cω ω udc  z4 = X = x1 x4 + i2d + u1q ∂X − C − L1C C L1C The relationship between the new control variable V and the original one U is as (10), where    2  r1 2ω 1+L1Cω ω  L r1(X)   x2 x3 i2q   f   C L1C C  Γ(X) =  r  =  − 2 − .  L 2 r (X)   2ω 1+L1Cω ω  f 2  x1 x4 + i2d  − C − L1C C Combining Equation (23) and Equation (10), the original control variable U can be expressed as

 2  L1C L1C 2ω 1+L1Cω ω  u1d = v1d [ i1q ucd i2q]  udc udc C L1C C − − 2 − (25)  L1C L1C 2ω 1+L1Cω ω  u1q = v1q [ i1d ucq + i2d] udc − udc − C − L1C C Electronics 2019, 8, 877 10 of 15

Through coordinate transformation, the original nonlinear system is transformed to two linear systems as follows: .  . " # " #" # " #  z1 0 1 z1 0  Z = . = +  d z 0 0 z v  2 2 1d  yd = z1 . (26)  . " # " #" # " #  z3 0 1 z3 0  Zq = . = +  z4 0 0 z4 v1q 11 of 16   yq = z3 Similarly, define the grid-side inductance current of the LCL filter as the state variables 𝑿= [𝑥 𝑥3.2.] =𝑖 State Feedback 𝑖 , the of Grid-Side filter capacitor Inductance voltage Subsystem as the input variables 𝑼=[𝑢 𝑢] =𝑢 𝑢 , and the gridSimilarly, side current define as the the grid-side output inductancevariable 𝒀=𝑖 current 𝑖 of the. LCL filter as the state variables T h iT h iT h iT = [ ] = = 𝜔𝑥 −𝑒=⁄𝐿 XThe modelx1 x2 of subsystemi2d i2q , the filtercan be capacitor written voltage as Equation as the input (2), where variables 𝑿U =u2d u2q ucdu ,cq 𝑮𝑿, = h iT −𝜔𝑥 −𝑒⁄𝐿 and the grid side current as the output variable Y = i2d i2q . 1𝐿⁄ 0 10L2 " # GX()==[] g () X g () X ωx e /L [𝑔𝑿 𝑔𝑿]= 12. ( ) 2 d 2 The model01𝐿 of subsystem⁄ can be written as Equation01 (2),L2 where X = − , ωx1 eq/L2 " # − − Then, the original control1 /variableL 0 U can be expressed as G(X) = [g (X) g (X)] = 2 . 1 2 0 1/L uLvLie2 =−ω + Then, the original control variable 22222Uddcan be expressed qd as  =+ω + (27) (uLvLie22222qq dq u = L v L ωi q + e 2d 2 2d − 2 2 d (27) u2q = L2v2q + L2ωi2d + eq 3.3. Parameters Design of Double Closed Loop Controller 3.3. Parameters Design of Double Closed Loop Controller In the reduced order feedback linearization method, the filter capacitor voltage is the input and In the reduced order feedback linearization method, the filter capacitor voltage is the input outputand of output the two of the reduced two reduced order order systems, systems, respecti respectively.vely. Then Then the doubledouble loop loop control control system system is is constructedconstructed that that the the outer outer control control l loopoop is thethe grid grid side side current current control control and theand inner the controlinner control loop is theloop is the filterfilter capacitor capacitor voltagevoltage control. control. The Thecontrol control inner inner loop loop is thethe filter filter capacitor capacitor voltage voltage control control and can and be designed can be asdesigned a unit negative as a unit negativefeedback feedback closed closed loop system. loop system. Considering Considerin the resonanceg the resonance suppression, suppression differential, feedback differential is needed. feedback is needed.Therefore, Therefore, a double a controldoubleloop control strategy loop can strategy be designed, can bein designed, which the ind -axiswhich system the d is-axis shown system in is shownFigure in Figure3. 3.

i1q 2ω ω i2q ed − C C

* u1d i2d LC1 1 i1d icd 1 ucd 1 i2d v2d L2 v1d udc udc sL1 sC sL2

i2q ed + ω2 − ω 1 LC1 L2 LC1

+ ks0 1

FigureFigure 3. Block 3. Block diagram diagram of ofthe the reduced reduced order order feedbackfeedback decoupling decoupling control control system. system. In Figure3, the VSI system is transformed to three series integration links. The outer control loopIn Figure is the grid-side3, the VSI inductance system is current transformed control, and to three the conventional series integration proportional links. integral The outer controller control loop isis compatible. the grid-side The innerinductance control loopcurrent is the contro filter capacitorl, and the voltage conventional control. In orderproportional to track highintegral controllerfrequency is compatible. ripple, a proportional The inner controller control can loop be adopted is the filter to improve capacitor the response voltage speed control. of the In system. order to trackDue high to thefrequency influence ripple, of the inner a proportional control loop on cont theroller original can system be structure,adopted theto outerimprove current the control response speedloop of the is no system. longer a Due precise to feedback.the influence A feedback of the terminne deviationr control appears. loop on However, the original compared system with structure, the the outer current control loop is no longer a precise feedback. A feedback term deviation appears. However, compared with the forward channel gain, the deviation, which can be equivalent to a feed forward interference, is smaller. Therefore, it can be compensated by a PI controller. Define the outer loop controller parameters as k2 and k3, the inner loop controller parameter as k1. Then, the control system can be represented as Figure 4.

ks0 * i + i2d 2d ks23 k 1 1 1 k1 s s s s

Figure 4. Equivalent block diagram of control system.

Open loop and closed loop transfer function of the system can be written as 11 of 16

Similarly, define the grid-side inductance current of the LCL filter as the state variables 𝑿= [𝑥 𝑥] =𝑖 𝑖 , the filter capacitor voltage as the input variables 𝑼=[𝑢 𝑢] =𝑢 𝑢 , and the grid side current as the output variable 𝒀=𝑖 𝑖 . 𝜔𝑥 −𝑒 ⁄𝐿 The model of subsystem can be written as Equation (2), where 𝑿 = , 𝑮𝑿 = −𝜔𝑥 −𝑒⁄𝐿 10L 1𝐿⁄ 0 ==[]2 [𝑔𝑿 𝑔𝑿]= GX() g12 () X g () X . 01𝐿⁄ 01L2 Then, the original control variable U can be expressed as =−ω + uLvLie22222dd qd  =+ω + (27) uLvLie22222qq dq

3.3. Parameters Design of Double Closed Loop Controller In the reduced order feedback linearization method, the filter capacitor voltage is the input and output of the two reduced order systems, respectively. Then the double loop control system is constructed that the outer control loop is the grid side current control and the inner control loop is the filter capacitor voltage control. The control inner loop is the filter capacitor voltage control and can be designed as a unit closed loop system. Considering the resonance suppression, differential feedback is needed. Therefore, a double control loop strategy can be designed, in which the d-axis system is shown in Figure 3.

i1q 2ω ω i2q ed − C C

* u1d i2d LC1 1 i1d icd 1 ucd 1 i2d v2d L2 v1d udc udc sL1 sC sL2

i2q ed + ω2 − ω 1 LC1 L2 LC1

+ ks0 1

Figure 3. Block diagram of the reduced order feedback decoupling control system.

In Figure 3, the VSI system is transformed to three series integration links. The outer control loop is the grid-side inductance current control, and the conventional proportional integral controller is compatible. The inner control loop is the filter capacitor voltage control. In order to track high frequency ripple, a proportional controller can be adopted to improve the response speedElectronics of the2019 system., 8, 877 Due to the influence of the inner control loop on the original system structure,11 of 15 the outer current control loop is no longer a precise feedback. A feedback term deviation appears. However,forward compared channel gain, with the deviation,the forward which channel can be gain equivalent, the deviation, to a feed forward which interference,can be equivalent is smaller. to a feedTherefore, forward itinterference, can be compensated is smaller. by aTherefor PI controller.e, it can be compensated by a PI controller. DefineDefine the the outer outer loop loop controller controller parametersparameters as ask2 kand2 andk3 ,k the3, the inner inner loop loop controller controller parameter parameter as k1. as k1. Then, thethe controlcontrol system system can can be be represented represented as Figureas Figure4. 4.

ks0 * i + i2d 2d ks23 k 1 1 1 k1 s s s s

FigureFigure 4. 4. EquivalentEquivalent block block diagram ofof controlcontrol system. system. Open loop and closed loop transfer function of the system can be written as Open loop and closed loop transfer function of the system can be written as

k1(k2s + k3) G (s) = (28) OL 2 2 s (s + k0k1s + k1)

k1(k2s + k3) G (s) = (29) CL 4 3 2 s + k0k1s + k1s + k1k2s + k1k3 In the double closed loop current control system shown in (29), the numerator order of the transfer function is quite different from the denominator order. The parameters coupling degree is low. Therefore, the parameters can be designed directly according to the system characteristics. As the parameters design method was analyzed before, this section does not repeat the description.

4. Simulations and Experiments

4.1. Simulation and Experimental Environment In order to verify the effectiveness of the proposed algorithm, the three-phase static var generator is used as the controlled object. Matlab/Simulink simulation software (2010b, MathWorks, Inc., Natick, Massachusetts 01760 USA) is used to carry out numerical simulation analysis of three-phase three-leg grid-connected inverter based on feedback linearization. IGBT model in MATLAB/Simulink simulation software is selected as switch, with internal resistance Ron = 1 mΩ, snubber resistance Rs = 500 kΩ, snubber capacitance Cs = inf. Furthermore, in order to verify the feasibility of the algorithm, a 50 kW prototype of three-phase three-leg inverter is built and the experiment is carried out. The Controller chip of the prototype is TMS320F28335 (Texas Instruments, Inc., Dallas, Texas 75243 USA). The IGBT module is SKM150GB12V (Semikron, Ltd., Nuremberg, Germany), with the switching frequency of 10 kHz. A photograph of the test rig is shown in Figure5. Based on the system parameters shown in Table1, the control parameters of the two feedback linearized systems are selected, respectively.

1. Based on Equations (17)–(20), selecting the cut-off frequency of the open-loop transfer function as 750 Hz, the single closed-loop control system parameters can be designed as: k3 = 5000, 7 4 k2 = 10 π, k1 = 200k2, k0 = 10 k2. At this time, the corresponding phase margin is Pm = 45◦, and the closed-loop bandwidth is 1 kHz. 2. Similarly, the parameters of the double closed loop control system can be designed as: k = 2 10 4, 0 × − k = 108, k = 5 103, k = 102k . Then, the corresponding open loop cut-off frequency is 680 Hz, 1 2 × 3 3 phase margin is Pm = 43◦, and closed-loop bandwidth is 1 kHz. 12 of 16

kksk()+ Gs()= 12 3 OL 22++ (28) s ()skksk01 1 kksk()+ Gs()= 12 3 CL 432++++ (29) s kks01 ks 1 kks 12 kk 13

In the double closed loop current control system shown in (29), the numerator order of the transfer function is quite different from the denominator order. The parameters coupling degree is low. Therefore, the parameters can be designed directly according to the system characteristics. As the parameters design method was analyzed before, this section does not repeat the description.

4. Simulations and Experiments

4.1. Simulation and Experimental Environment In order to verify the effectiveness of the proposed algorithm, the three-phase static var generator is used as the controlled object. Matlab/Simulink simulation software (2010b, MathWorks, Inc., Natick, Massachusetts 01760 USA) is used to carry out numerical simulation analysis of three-phase three-leg grid-connected inverter based on feedback linearization. IGBT model in MATLAB/Simulink simulation software is selected as switch, with internal resistance Ron = 1 mΩ, snubber resistance Rs = 500 kΩ, snubber capacitance Cs = inf. Furthermore, in order to verify the feasibility of the algorithm, a 50 kW prototype of three- phase three-leg inverter is built and the experiment is carried out. The Controller chip of the prototype is TMS320F28335 (Texas Instruments, Inc., Dallas, Texas 75243 USA). The IGBT module is SKM150GB12VElectronics 2019, 8, 877 (Semikron, Ltd., Nuremberg, Germany), with the switching frequency of 10 kHz12. of 15 A photograph of the test rig is shown in Figure 5.

Oscilloscope Current probes Inverter

Inverter Inverter Switching power supply IGBTs and drivers Control board 13 of 16 Load resistance Inverter Grid VoltageInverter ug 380 V Grid-side inductor L2 0.2 mH Inverter side L1 0.3 mH

inductor Inverter Filter capacitor C 20 uF Switching frequencyInductance f 10 kHz

1. Based on Equations (17)–(20), selecting the cut-off frequency of the open-loop transfer function Figure 5. Photograph of the experimental test rig. as 750 Hz, the single Figureclosed-loop 5. Photograph control of system the experimental parameters test rig.can be designed as: k3=5000, k2=107π, k1=200k2, k0=104k2,,,. At this time, the corresponding phase margin is Pm=45°, and the Based on the system parametersTable shown 1. Parameters in Table for 1, thethe system. control parameters of the two feedback closed-loop bandwidth is 1 kHz. linearized systems are selected,Variables respectively. Symbols Value 2. Similarly, the parameters of the double closed loop control system can be designed as: k0=2*10-4, DC Bus Voltage u 650 V =×Table− 41. Parameters= 8 for=× dcthe system.3 = 2 Gridk0 210 Voltage k1 10 k2 u510g kk3310380 V k1=108, k2=5*103, k3=102k3 , , , . Then, the corresponding Grid-side inductor L 0.2 mH Variables Symbols2 Value =  Inverter side inductor L1 0.3 mHPm 43 open loop cut-off frequency is 680 Hz, phase dcmargin is Pm=43° , and closed-loop DCFilter Bus capacitor Voltage u C 65020 V uF bandwidth is 1 kHz. Switching frequency f 10 kHz The corresponding frequency domain characteristics of two control systems are shown in FigureThe 6. corresponding frequency domain characteristics of two control systems are shown in Figure6.

Bode Diagram Bode Diagram 60 60 Open Loop Open Loop 40 Closed Loop 40 Closed Loop 20 20 0 0 -20 -20 Magnitude (dB) Magnitude

Magnitude (dB) Magnitude -40 -40 -60 -60 0 90

0 -90

-90 -180 Phase (deg) Phase

Phase (deg) Phase -180 -270 -270 1 2 3 4 1 2 3 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) (a) (b)

FigureFigure 6. 6. BodeBode diagrams diagrams of of the the closed-loop closed-loop control control systems: systems: (a)( aSingle) Single closed-loop closed-loop control control system, system, (b) double(b) double closed-loop closed-loop control control system. system.

4.2.4.2. Steady Steady State State Control Control Performance Performance Setting VSI output current as i = 50 A, the three-phase output current on the basis of two control Setting VSI output current as2 i2 = 50 A, the three-phase output current on the basis of two controlmethods methods is shown is inshown Figure in7 Figure. 7.

Ia Ia 100 100 Ib Ib Ic Ic 50 50

/A 0 2 /A 0 I 2 I

-50 -50

-100 -100

0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 t/s t/s (a) (b)

Figure 7. Steady-state current waveforms of voltage source inverters (VSI) based on two control schemes: (a) Output current of the single closed-loop control system, (b) output current of the double closed-loop control system. 13 of 16

Grid Voltage ug 380 V Grid-side inductor L2 0.2 mH Inverter side L1 0.3 mH inductor Filter capacitor C 20 uF Switching frequency f 10 kHz

1. Based on Equations (17)–(20), selecting the cut-off frequency of the open-loop transfer function as 750 Hz, the single closed-loop control system parameters can be designed as: k3=5000, k2=107π, k1=200k2, k0=104k2,,,. At this time, the corresponding phase margin is Pm=45°, and the closed-loop bandwidth is 1 kHz.

2. Similarly, the parameters of the double closed loop control system can be designed as: k0=2*10-4, =× −4 = 8 =× 3 = 2 k0 210 k1 10 k2 510 kk3310 k1=108, k2=5*103, k3=102k3 , , , . Then, the corresponding =  Pm 43 open loop cut-off frequency is 680 Hz, phase margin is Pm=43° , and closed-loop bandwidth is 1 kHz. The corresponding frequency domain characteristics of two control systems are shown in Figure 6.

Bode Diagram Bode Diagram 60 60 Open Loop Open Loop 40 Closed Loop 40 Closed Loop 20 20 0 0 -20 -20 Magnitude (dB) Magnitude

Magnitude (dB) Magnitude -40 -40 -60 -60 0 90

0 -90

-90 -180 Phase (deg) Phase

Phase (deg) Phase -180 -270 -270 1 2 3 4 1 2 3 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) (a) (b)

Figure 6. Bode diagrams of the closed-loop control systems: (a) Single closed-loop control system, (b) double closed-loop control system.

4.2. Steady State Control Performance

Setting VSI output current as i2 = 50 A, the three-phase output current on the basis of two Electronics 2019, 8, 877 13 of 15 control methods is shown in Figure 7.

Ia Ia 100 100 Ib Ib Ic Ic 50 50

/A 0 2 /A 0 I 2 I

-50 -50

-100 -100

0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 t/s t/s (a) (b)

FigureFigure 7. Steady-state 7. Steady-state current current waveforms waveforms of of voltage voltage source invertersinverters (VSI) (VSI) based based on on two two control control schemes: (a) Output current of the single closed-loop control system, (b) output current of the double schemes: (a) Output current of the single closed-loop control system, (b) output current of the14 of 16 doubleclosed-loop closed-loop control control system. system. According to Figure 7, both the two inverter systems can track the reference current accurately According to Figure7, both the two inverter systems can track the reference current accurately with small current distortion. Compared to single closed loop control, the output ripple of double with small current distortion. Compared to single closed loop control, the output ripple of double closed loop control system is smaller. closed loop control system is smaller. The prototype is adopted to carry out the experiment. The output current of the system based The prototype is adopted to carry out the experiment. The output current of the system based on ontwo two kinds kinds of control of control methods methods is shown is in shown Figure8 .in The Figure output current8. The Totaloutput Harmonic current Distortion Total Harmonic (THD) Distortionwith the two(THD) control with methods the two are 4.36%control and methods 1.57%, respectively. are 4.36% Theand current 1.57%, ripple respectively. of single closed The loopcurrent ripplecontrol of systemsingle basedclosed on loop feedback control linearization system based is larger. on Thefeedback main reason linearization is that the is control larger. system The main is reasonhighly is dependent that the oncontrol the accuracy system ofis the highly system dependent model. However, on the the accuracy sampling of and the control system delays, model. However,which exist the insampling real systems, and control cause the delays, inaccuracy which of exist the feedbackin real systems, signal. Incause addition, the inaccuracy the nonlinear of the feedbackmagnetization signal.curve In addition, of the inductance the nonlinear and the equivalentmagnetization series curve resistance of the of theinductance capacitance and also the equivalentaffect the modelseries accuracy.resistance Itof reduces the capacitance the damping also effect affect of the th resonancee model pointaccuracy. and increasesIt reduces the the dampingcurrent ripple.effect of The the reduced resonance order point double and closed increases loop controllerthe current takes ripple. the filterThe capacitorreduced order voltage double as closedthe intermediate loop controller variable. takes To the some filter extent, capacitor the measure voltage reducesas the intermediate the coupling relationshipvariable. To among some extent, the thecontrol measure variables reduces and the improves coupling the robustnessrelationship of theamong system. the The control steady variables state control and performance improves the robustnessis better. of the system. The steady state control performance is better.

(a) (b)

FigureFigure 8. 8.ExperimentalExperimental steady-state steady-state current current waveforms:waveforms: (a(a)) Output Output current current of of the the single single closed-loop closed-loop controlcontrol experiment, experiment, (b (b) )output output current current ofof thethe doubledouble closed-loop closed-loop control control experiment. experiment. 4.3. Dynamic Control Performance 4.3. Dynamic Control Performance In this paper, a dynamic test is carried out in the form of virtual load. The grid-connected current In this paper, a dynamic test is carried out in the form of virtual load. The grid-connected is set as 25 A at the initial time, and then doubles at 0.3 s. The output current of the system is shown in currentFigure is9. set as 25 A at the initial time, and then doubles at 0.3 s. The output current of the system is shownWhen in Figure the load 9. current changes, the output current of the two systems fluctuates with little overshoot.When the After load a brief current transient changes, process, the the output system reachescurrent aof new the steady two statesystems within fluctuates one fundamental with little overshoot.period. Compared After a tobrief the directtransient feedback process, linearization the system control, reaches the reduced a new order steady feedback state linearization within one fundamental period. Compared to the direct feedback linearization control, the reduced order feedback linearization control lacks one closed loop zero, which reduces the response speed. However, the transition process is relatively smooth. Therefore, it can be seen that the two control schemes both have fast response and small overshoot.

100 100 Ia Ia Ib Ib Ic Ic 50 Iref(amp) 50 Iref(amp)

/A 0 2 /A 0 I 2 I

-50 -50

-100 -100 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 t/s t/s (a) (b) 14 of 16

According to Figure 7, both the two inverter systems can track the reference current accurately with small current distortion. Compared to single closed loop control, the output ripple of double closed loop control system is smaller. The prototype is adopted to carry out the experiment. The output current of the system based on two kinds of control methods is shown in Figure 8. The output current Total Harmonic Distortion (THD) with the two control methods are 4.36% and 1.57%, respectively. The current ripple of single closed loop control system based on feedback linearization is larger. The main reason is that the control system is highly dependent on the accuracy of the system model. However, the sampling and control delays, which exist in real systems, cause the inaccuracy of the feedback signal. In addition, the nonlinear magnetization curve of the inductance and the equivalent series resistance of the capacitance also affect the model accuracy. It reduces the damping effect of the resonance point and increases the current ripple. The reduced order double closed loop controller takes the filter capacitor voltage as the intermediate variable. To some extent, the measure reduces the coupling relationship among the control variables and improves the robustness of the system. The steady state control performance is better.

(a) (b)

Figure 8. Experimental steady-state current waveforms: (a) Output current of the single closed-loop control experiment, (b) output current of the double closed-loop control experiment.

4.3. Dynamic Control Performance In this paper, a dynamic test is carried out in the form of virtual load. The grid-connected current is set as 25 A at the initial time, and then doubles at 0.3 s. The output current of the system is shown in Figure 9. When the load current changes, the output current of the two systems fluctuates with little overshoot.Electronics 2019 After, 8, 877 a brief transient process, the system reaches a new steady state within14 ofone 15 fundamental period. Compared to the direct feedback linearization control, the reduced order feedbackcontrol lackslinearization one closed control loop zero, lacks which one reduces closed the loop response zero, speed.which However, reduces thethe transition response process speed. However,is relatively the smooth. transition Therefore, process it is can relatively be seen thatsmooth. the two Therefore, control schemes it can be both seen have that fast the response two control and schemessmall overshoot. both have fast response and small overshoot.

100 100 Ia Ia Ib Ib Ic Ic 50 Iref(amp) 50 Iref(amp)

/A 0 2 /A 0 I 2 I

-50 -50

-100 -100 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 t/s t/s 15 of 16 15 of 16 (a) (b)

(c()c ) ((dd))

FigureFigure 9. 9.9.Dynamic DynamicDynamic current currentcurrent waveforms waveformswaveforms of of VSI VSI basedbased based onon on two two two control controlcontrol schemes: schemes:schemes: (a )((a Output)) Output current current of ofof thethe single single closed-loop closed-loop control controlcontrol system, system,system, (b ((bb) )output) outputoutput current currentcurrent of ofof the thethe double doubledouble closed-loop closed-loopclosed-loop controlcontrolcontrol system,system, ((cc)) partial(partialc) partial enlarged enlarged enlarged detail detail detail of of (a of(),a ), ((ad ),()d (partial)d partial) partial enlarged enlarged enlarged detail detail detail of of of ( b(b ().b). ).

TheThe experimental experimentalexperimental conditions conditions conditions areare are the the the same same same as as thoseas th thoseose of theof of the simulation.the simulation.simulation. The experimentalTheThe experimental results resultsresults are shown in Figure 10. Similar to the simulation results, it is shown that both of the two control schemes areare shown shown in inFigure Figure 10. 10. Similar Similar to to the the simulation simulation results, results, itit isis shownshown thatthat both of the two controlcontrol have good dynamic performance. schemesschemes have have good good dynamic dynamic performance. performance.

(a()a ) ((bb))

FigureFigure 10. 10.10. Experimental ExperimentalExperimental dynamic dynamicdynamic current current current waveforms: waveforms: (( aa()a) Output)Output Output current currentcurrent of ofof the thethe single singlesingle closed-loop closed-loop controlcontrol experiment, experiment,experiment, (b ( b()b )output) output output current current current of of thethe the doubledouble double closed-loop closed-loop closed-loop control control control experiment. experiment.experiment. 5. Conclusions 5. Conclusion5. Conclusion In orderorder toto solvesolve thethe problem problem of of poor poor global global control control performance performance of of the the controller controller designed designed by by In order to solve the problem of poor global control performance of the controller designed by small signalsignal modelingmodeling andand harmonicharmonic linearization, linearization, two two universal universal controller controller design design methods methods based based small signal modeling and harmonic linearization, two universal controller design methods based on statestate feedbackfeedback linearization linearization are are proposed proposed for thefor LCL the typeLCL VSI. type Relation VSI. Relation degree ofdegree the VSI of system the VSI onand state that feedback of the reduced linearization order model are areproposed analyzed, for respectively. the LCL type Subsequently, VSI. Relation the coupling degree matrixesof the VSI system and that of the reduced order model are analyzed, respectively. Subsequently, the coupling systemare designed and that by of adoptingthe reduced the Lieorder derivative. model are Through analyzed, the respectively. nonlinear transformation, Subsequently, the the original coupling matrixes are designed by adopting the Lie derivative. Through the nonlinear transformation, the matrixessystem isare mapped designed to the by Brunovsky adopting standardthe Lie type,deriva andtive. the Through new system the linear nonlinear controller transformation, can be used to the original system is mapped to the Brunovsky standard type, and the new system linear controller original system is mapped to the Brunovsky standard type, and the new system linear controller can be used to complete the control of the original complex system. Then, the parameters design can be used to complete the control of the original complex system. Then, the parameters design ideas of two kinds of controllers are given. The proposed single closed-loop current controller has a ideassimple of two structure, kinds ofbut controllers it is highly are dependent given. The on proposed the exact single model closed-loop of the system. current The controller double closed has a simpleloop structure,controller butsets itthe is highlyfilter capacitor dependent voltage on the as exact the innermodel loop of the control system. variable, The double by which closed a loopreduced controller order sets feedback the filter linearization capacitor voltagecontrol canas thebe innerachieved loop step control by step,variable, to improveby which the a reducedadaptability order and feedback robustness linearization of the system. control The cansimu belation achieved and experimental step by step, results to showimprove that thethe adaptabilitytwo controllers and robustnesshave good steady-stateof the system. control The pesimurformancelation and and experimental dynamic resp resultsonse performance. show that the In twopractical controllers application, have good the steady-staterobustness ofcontrol linear pe corformancentrol system and based dynamic on reduced-order response performance. feedback Inis practicalbetter. application, the robustness of linear control system based on reduced-order feedback is better. Author Contributions: L.Y. wrote the paper and designed the control method; C.F. contributed to the Authorconception Contributions: of the study L.Y. and wrote designed the paper the simulationand designed model; the J.L.control helped method; to perform C.F. contributedthe analysis to with the conceptionconstructive of discussions.the study and designed the simulation model; J.L. helped to perform the analysis with constructive discussions. Acknowledgments: This work is supported by the National Natural Science Foundation of China (Grant No. Acknowledgments:51607179) and the FundamentalThis work is Researchsupported Funds by the for National the Central Natu Universitiesral Science (2017QNB01). Foundation of China (Grant No. 51607179) and the Fundamental Research Funds for the Central Universities (2017QNB01). Conflicts of Interest: No potential conflict of interest was reported by the authors. Conflicts of Interest: No potential conflict of interest was reported by the authors. Electronics 2019, 8, 877 15 of 15 complete the control of the original complex system. Then, the parameters design ideas of two kinds of controllers are given. The proposed single closed-loop current controller has a simple structure, but it is highly dependent on the exact model of the system. The double closed loop controller sets the filter capacitor voltage as the inner loop control variable, by which a reduced order feedback linearization control can be achieved step by step, to improve the adaptability and robustness of the system. The simulation and experimental results show that the two controllers have good steady-state control performance and dynamic response performance. In practical application, the robustness of linear control system based on reduced-order feedback is better.

Author Contributions: L.Y. wrote the paper and designed the control method; C.F. contributed to the conception of the study and designed the simulation model; J.L. helped to perform the analysis with constructive discussions. Acknowledgments: This work is supported by the National Natural Science Foundation of China (Grant No. 51607179) and the Fundamental Research Funds for the Central Universities (2017QNB01). Conflicts of Interest: No potential conflict of interest was reported by the authors.

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