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Stabilization Design of Three-Phase LCL-Filtered Grid-Connected Inverter Using IDA-PBC Controller

Stabilization Design of Three-Phase LCL-Filtered Grid-Connected Inverter Using IDA-PBC Controller Hindawi International Transactions on Electrical Energy Systems Volume 2022, Article ID 7515321, 14 pages https://doi.org/10.1155/2022/7515321 Research Article StabilizationDesignofThree-PhaseLCL-FilteredGrid-Connected Inverter Using IDA-PBC Controller 1 1 1 1 1 Min Huang , Zhicheng Zhang , Fan Chen , Weimin Wu , Zhilei Yao , and Frede Blaabjerg Department of Electrical Engineering, Shanghai Maritime University, Shanghai 201306, China Department of Energy Technology, Aalborg University, Aalborg 9220, Denmark Correspondence should be addressed to Min Huang; minhuang@shmtu.edu.cn Received 7 December 2021; Revised 24 February 2022; Accepted 15 March 2022; Published 15 April 2022 Academic Editor: Tomislav Capuder Copyright © 2022 Min Huang et al. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. With the aim of improving the stability of renewable energy system with high permeability in the weak grid, a modified passivity- based control based on interconnection and damping assignment (IDA) is presented for LCL-filtered grid-connected inverters to eliminate the interactive resonances. (e object is modeled in the form of port-controller Hamiltonian, including the partial differential of stored energy function. On the premise of ensuring Lyapunov stability, a more flexible interconnection matrix design method is applied to simplify the design process. (e closed-loop stability is ensured with the selected Hamiltonian energy function at the desired balanced point. Moreover, a step-by-step design process for damping gains is provided to guarantee stable operation and fast dynamic response under variation of complex grid impedance. With the designed injected damping pa- rameters, considering the effect of delay, it is possible for the real part of the inverter output admittance to be positive within the switching frequency. (e performance and robustness of the proposed method for LCL-filtered inverter system are validated via simulated and experimental results under both unbalanced and balanced grid conditions. stability with the grid [5]. During actual operation, grid 1. Introduction impedance is varying, resulting in stimulated resonances for In renewable energy generation system, the grid-connected the stable inverter system during independent operation and inverter has become one of the core parts to interface grid even undesirable tripping. Recently, the concept of passivity [1]. Its output current harmonic distortion is an important theory was applied to access the stability of grid-connected technical indicator; thus, LCL filter with low-pass charac- inverters in frequency domain, and many research works teristics is usually connected to inject high quality current. have been proposed to extend nonnegative real part of in- LCL filter has better filtering effect, smaller system volume, verter output admittance [5–10]. It indicates that the res- and lower loss at the same cost compared with L filter [2]. onance will not be stimulated if the interactive point falls in the passive region. Most control methods mainly consider However, as a result of inherent resonance of high-order filter, the circuit is prone to oscillating or even unstable passive regions up to the Nyquist frequency. However, outputs. [3] (erefore, extra requirements are put forward passivity analysis and instability of inverter admittance for the controller design for LCL-filtered inverter system. above the Nyquist frequency should also be considered [8]. Additionally, 1-beat or 1.5-beat delay could be brought by Currently, most closed-loop control strategies rely on the discrete control and pulse width modulation [4]. (e control classic linear control theory, which can be used to accurately delay will typically reduce the phase margin as well as design the controller parameters and evaluate the influence complicate the design of the controller. of the delay. (ese control methods are however difficult to Apart from the stability of the inverter system itself, fully eliminate interactive resonances under complex weak another concern of the controller design is the interactive grid, and the traditional proportional integral control does 2 International Transactions on Electrical Energy Systems demonstrated by simulations and experimental results. not have sufficient disturbances rejection capability. Com- pared with the classical linear control theory, the modern Furthermore, the study found that IDA-PBC also performed well in unbalanced grid. Section 7 concludes the study. nonlinear control theory has better robustness and per- formance, such as sliding mode control [11, 12], predictive control [13, 14], and passivity-based control (PBC) [15–26]. 2. System Description and PCH Modeling PBC has proven useful in the design of robust controllers for railway systems [19, 20], renewable generation systems [21], 2.1. Description of the System. Figure 1 depicts an inverter etc. In terms of the PBC design process, energy formation with a power supply voltage U and is filtered by a LCL filter dc and damping injection are considered to be of obvious including inverter-side inductor L , filter capacitor C, and physical significance [17], and two branches are proposed by grid-side inductor L . (e parasitic resistances of L and L 2 1 2 the Euler–Lagrange (EL) model and the port-controller are represented by R and R , respectively. (e voltage v is 1 2 pcc Hamiltonian (PCH) model, called EL-PBC and IDA-PBC, used for the synchronization of the grid current. Addi- respectively. [22–26] (e modeling process of the EL-PBC tionally, the inverter-side current i , the capacitor voltage u , 1 C method is simpler, but IDA-PBC offers more flexibility in and the grid-side current i should also be measured to the type of damping interconnection and the modeling control the injected current as well as suppress the reso- process. Moreover, for the IDA-PBC, fast convergence speed nances. It should be noted that the state observer or Kalman of the error energy function can be determined by calcu- filter can be used to reduce the number of required sensors lating the derivative of the closed-loop energy function and by estimating state variables [9, 11]. (e equivalent grid the injected damping [23]. For EL-PBC, when the error is reactance of L and C is connected to the grid at the point of g g large, the convergence speed is fast, and injected damping common coupling (PCC), which can take a wide range of plays the main role. In contrast, the convergence speed values. (erefore, the controller must be robust enough to becomes slower and the role of injected damping becomes withstand grid disturbance. Table 1 summarizes basic system weak when the error is small. Towards this end, IDA-PBC parameters for analysis. has been adopted in this study so as to achieve better di ⎧ ⎪ 1k performance under complex grid conditions. ⎪ ⎪ L + R i + u � u , 1 1 1k Ck ik dt However, it is often overlooked that time delay can affect ⎪ IDA-PBC [25]. As to the design of damping matrix of IDA- PBC, few research studies analyze the specific design du Ck (1) C + i − i � 0, 2k 1k guideline and the try-and-error method is usually adopted. dt (erefore, for the purpose of assessing the stability of weak grid, the detailed design of digital controlled IDA-PBC is di 2k L + R i − u � −v . necessary. A brief summary of the main contributions is as 2 2 2k Ck pcck dt follows. In accordance with Figure 1, the equations for a three- (1) On the premise of ensuring Lyapunov stability, to phase LCL filtered inverter system can be derived from simplify the design process for PCH model-based Kirchhoff’s law in a stationary abc frame as follows:where PBC, a more flexible interconnection matrix design k stands for the cases in abc coordinates. (rough the method is used. rotation transformation, from (1), the mathematical (2) (is study demonstrates how to select the injected model of the grid-connected inverter in d-q coordinates damping considering the effect of control delay using appears as frequency-domain passivity theory based on linear di control design. ⎪ 1d L � u − R i + ωL i − u , 1 id 1 1d 1 1q Cd dt (3) With the proposed controller parameters’ design method, the nonnegative real part of the LCL-filtered di ⎪ 1q inverter output admittance can be achieved within L � u − R i − ωL i − u , 1 iq 1 1q 1 1d Cq switching frequency. If the passivity of subsystem is dt ensured, then the stability of the whole interlinked du ⎧ ⎪ Cd system in parallel form is guaranteed regardless of C � i − i + ωCu , 1d 2d Cq dt grid impedance. (2) Moreover, in Section 2, a brief description of the LCL- du ⎪ Cq filtered inverter system is given, followed by the design of C � i − i − ωCu , 1q 2q Cd dt IDA-PBC methodology in Section 3. Instead of the original antisymmetric matrix, the interconnection matrix is sim- di 2d ⎧ ⎪ plified by using the upper triangular matrix with zero corner L � u − R i + ωL i − v , ⎪ 2 Cd 2 2d 2 2q pccd ⎪ dt elements. In order to precisely design the injected damping gains, in Section 4, frequency-domain passivity theory is ⎪ di applied with the derived impedance model of IDA-PBC. In ⎪ 2q L � u − R i − ωL i − v . 2 Cq 2 2q 2 2d pccq Section 5 and Section 6, system performance is dt International Transactions on Electrical Energy Systems 3 Table 1: System parameters. LCL-filter PCC L , R L , R g Symbols Components Value 1 1 2 2 U DC bus voltage 350 V dc 1 C U Grid voltage 110V (RMS) dc f Grid frequency 50 Hz i pcc i u 2 Grid 1 C impedance L Inverter-side inductor 1.2 mH C Filter capacitor 6 uF L Grid-side inductor 1.2 mH SV IDA-PBC PWM f Sampling frequency 10 kHz f Switching frequency 10 kHz sw Figure 1: Structure of LCL-filtered three-phase grid-connected inverter. 3. IDA-PBC Controller Design 2.2. PCH Modeling. In order to construct a robust controller (e primary goal of the controller is to find the Hamiltonian for the above system, a PCH model with dissipation can be energy storage function H (x), interconnection matrix J (x), a a used [24]. It is possible to express the system’s PCH model as and damping matrix R (x), which satisfy the following follows: equation with a balance point at x : zH(x) (3) x _ � [J(x) − R(x)] + g(x)u, zH(x) zx 􏼂J (x) − R (x)􏼃K(x) � −􏼂J (x) − R (x)􏼃 + g(x)u, d d a a zx where J(x) represents the system interconnection prop- (8) erties and normally satisfy formula J(x) � -J(x) , R(x) where K(x) � [zH (x)/zx], J (x) � J(x) + J (x), indicates the system dissipation characteristics, and a d a R (x) � R(x) + R (x), and H (x) is the undetermined H(x) is the Hamiltonian function; u indicates the ex- d a function. change of energy between the outside world and the Moreover, in order to ensure x is locally stable equi- system. (e coefficient matrix g(x) stands for the inter- librium point, the following conditions should also be sat- connections. (e following is a definition of system isfied [15]. variables: (1) In traditional design, strict structure preservation is x � 􏼂 x x x x x x 􏼃 1 2 3 4 5 6 provided (4) L i L i Cu Cu L i L i � 􏽨 􏽩 . 1 1d 1 1q Cd Cq 2 2d 2 2q T ⎧ ⎨ J (x) � J(x) + J (x) � −􏼂J(x) + J (x)􏼃 , d a a (9) (e following is the expression for each of the R (x) � R(x) + R (x) � 􏼂R(x) + R (x)􏼃 . d a a vectors and matrices for the LCL-filter inverter according to (2): (2) K(x) is the integrated gradient of a scalar function: 0 ωL −1 0 0 0 1 zK(x) zK(x) ⎡ ⎢ ⎤ ⎥ (10) ⎢ ⎥ � 􏼠 􏼡 . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ zx zx ⎢ −ωL 0 0 −1 0 0 ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 0 0 ωC −1 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (3) Expect the balance x ; the desired dynamics is ⎢ ⎥ J � ⎢ ⎥, ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 1 −ωC 0 0 −1 ⎥ ⎢ ⎥ achieved if K(x) is satisfied: ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 1 0 0 ωL ⎥ ⎢ ⎥ zH(x) ⎢ 2 ⎥ ⎢ ⎥ ∗ ⎣ ⎦ (11) K x 􏼁 � − | ∗ , (5) x�x zx 0 0 0 1 −ωL 0 1 0 0 0 0 0 H (x) � H(x) + H (x). (12) d a ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 1 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ g(x) � ⎢ ⎥ , ⎢ ⎥ (4) At the balance point, K(x) should satisfy the fol- ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 −1 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ lowing function based on the Lyapunov stability: 0 0 0 0 0 −1 zK(x) z H(x) u � 􏽨u , u , v , v 􏽩 , (13) d q pccd pccq | > − | . ∗ ∗ x�x x�x (6) 2 zx zx R � diag􏼂 R R 0 0 R R 􏼃. 1 1 2 2 For EL-PBC, J(x) is fixed and the damping is injected (e stored energy in the LCL filter can be expressed as with constructed energy function. For IDA-PBC, it is more flexible to design J (x) and R (x) corresponding to desired 2 2 2 2 2 2 a a x + x x + x x + x 1 2 3 4 5 6 energy function. At the stable equilibrium point x , the (7) H(x) � + + . 2L 2C 2L 1 2 closed-loop system based on PCH is described as [22] 4 International Transactions on Electrical Energy Systems and x can gradually reach the equilibrium points, which are zH (x) 4 x _ � J (x) − R (x) . (14) 􏼂 􏼃 d d defined as follows: zx ∗ ∗ ∗ ∗ ∗ ∗ (17) At the desired x , the tracking error of selected Ham- x∗ � 􏽨L i , L i , Cu , Cu , L i , L i 􏽩 . 1 1d 1 1d Cd Cq 2 2d 2 2q iltonian function H (x) should be minimum. (e derivation (e derivative of H (x) can be defined as K(s) and of the closed-loop energy function H (x) is depicted as d corresponding components of K(s) can be viewed as follows: zH (x) _ T H (x) � 􏼢 􏼣 x _ (18) d K(x) � 􏼂k (x), k (x), k (x), k (x), k (x), k (x)􏼃 , 1 2 3 4 5 6 zx (15) ∗ ∗ x x T ⎧ ⎪ 1 2 k (x) � − , k (x) � − , zH (x) zH (x) ⎪ 1 2 d d � − R (x) − J (x) . ⎪ L L 􏼢 􏼣 􏼂 􏼃 1 1 d d ⎪ zx zx ∗ ∗ x x 3 4 (19) Based on Lyapunov’s second criterion, the derivative of k (x) � − , k (x) � − , ⎪ 3 4 C C H (x) of the closed-loop energy function must be negative to ∗ ∗ attain an asymptotically stable equilibrium point. Conven- ⎪ x x 5 6 ⎩ (x) � − , k (x) � − , 5 6 tional, J is designed as positive definite symmetric matrix L L 2 2 based on structure preservation in (9). A more flexible method is proposed to design J in this study. (e derivative When J , R , and K(x) are substituted to (1), reference d a a of the energy function can be less than zero if [R -J ] is a values for the state variables of the system can be found in d d positive definite matrix. So, if injected damping R is pos- (20) which determined the passive controller u. As a result of itive, J can be designed as a lower triangular matrix or an (20), as shown in Figure 2, the cascaded three loops con- upper triangular matrix with zero diagonal elements to troller can be depicted. It is worth mention that “L ,” “L ,” 1e 2e simplify the design procedure, and it is also capable of “C ,” “R ,” and “R ” are the initial filter parameters given in e 1e 2e ensuring that H (x)< 0. So, J , J , and R are defined as the controller, and the subscript “e” is used to differentiate a d a with actual parameters of thr physical filter. 0 −ωL 1 0 0 0 ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ ∗ ∗ ⎢ ⎥ ⎢ ωL 0 0 1 0 0 ⎥ ⎢ ⎥ u � u − ωL i + r i − i 􏼁 + R i , ⎢ ⎥ ⎢ 1 ⎥ ⎧ ⎪ cd 1e 1q 1 1d 1e ⎢ ⎥ d 1d 1d ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ∗ ∗ ∗ ⎢ ⎥ ⎢ 0 0 0 −ωC 1 0 ⎥ ⎪ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ u � u + ωL i + r 􏼐i − i 􏼑 + R i , ⎢ ⎥ ⎢ ⎥ q cq 1e 1d 1 1q 1q 1e 1q ⎢ ⎥ ⎪ ⎢ ⎥ J � ⎥, ⎢ ⎥ ⎪ a ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ 0 0 ωC 0 0 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ ∗ 2d ∗ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ i � − ωC u + r u − u , ⎢ ⎥ 􏼁 ⎢ ⎥ ⎨ ⎢ ⎥ 1d i e Cq 2 Cd Cd ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (20) ⎢ 0 0 0 0 0 −ωL ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎪ ⎣ ⎦ 2q ∗ ∗ i � + ωC u + r 􏼐u − u 􏼑, ⎪ 1q i e Cd 2 Cq Cq 0 0 0 0 ωL 0 ∗ ∗ u � v − ωL i + r i − i 􏼁 + R i , Cd pccd 2e 2q 3 2d 2d 2e 2d 0 0 0 0 0 0 ⎪ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎩ ⎢ ⎥ ∗ ∗ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ u � v + ωL i + r i − i + R i . ⎢ ⎥ 􏼐 􏼑 ⎢ ⎥ ⎢ 0 0 0 0 0 0 ⎥ Cq pccq 2e 2d 3 2q 2q 2e 2q ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ By using (11) and (12), it is possible to calculate the ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ J ⎢ ⎥, R � ⎢ ⎥ � diag􏼂 r r r r r r 􏼃. ⎢ ⎥ d ⎢ ⎥ a 1 1 2 2 3 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ amount of energy injected into the system by the controller: ⎢ ⎥ ⎢ ⎥ ⎢ 0 1 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎥ ⎢ 0 0 1 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x x x x x x ⎣ ⎦ 1 2 3 4 5 6 H (x) � − x − x − x − x − x − x . a 1 2 3 4 5 6 0 0 0 1 0 0 L L C C L L 1 1 2 2 (16) (21) (e elements of R should be positive. (rough the Accordingly, the closed-loop Hamiltonian energy ∗ ∗ control states x ⟶x and x ⟶x , the states x , x , x , function for this system is as follows: 5 5 6 6 1 2 3 ∗ ∗ ∗ ∗ ∗ ∗ 2 2 2 2 2 2 x x x x x x x + x x + x x + x 1 2 3 4 5 6 1 2 3 4 5 6 (22) H (x) � − x − x − x − x − x − x + + + . d 1 2 3 4 5 6 L L C C L L 2L 2C 2L 1 1 2 2 1 2 1 1 1 1 1 1 z H (x) (23) | ∗ � diag􏼢 􏼣> 0. At the balanced point x , zH (x)/zx is equal to 0, and x�x L L C C L L zx 1 1 2 2 the Hessian matrix of H (x) shown in (23) meets the re- quirement of (13): Hence, H (x) has the minimum value at x . Furthermore, d International Transactions on Electrical Energy Systems 5 LCL-filter Controller u u i v pccd i Cd Cd 2d pccd 2d 2e * 1e i + u u 1d + d d i * - -- 2d u * i Cd Cd 2d + + i 1 1 1 1d r r r G (s) 3 2 1 d sL +R sC sL +R 1 1 - - - 2 2 - - - + + Cd 2d ωL ωL ωC 1d ωL ωL 2e 1e ωC e 1 2 Cq i ωL 1q ωL ωL ωL 2q ωC ωC 2e 1e 1 2 e * u u q q + + * - -- i - - - 2q 1 1 1 r r r G (s) 3 2 1 d * + sL +R + * sC sL +R + 1 1 u i 2 2 u Cq 1q Cq + + - -- i + i 2q 1q R R 2e 1e u v v i Cq pccq Cq i pccq 2q 2q Figure 2: Proposed IDA-PBC strategies for LCL-filtered grid-connected inverter. is dissipative, this also implies that it is passive. In order to zH (x) H(x) � x _ 􏼢 􏼣 select the values of the injected damping coefficients, Section zx 4 will analyze the passivity of the closed-loop output ad- (24) mittance using the impedance model. zH (x) zH (x) d d � −􏼢 􏼣 􏼂R (x) − J (x)􏼃 ≤ 0. d d zx zx 4. Impedance Model and Stability Oriented Design And apart from x , there are no solutions of x (t) which can stay in the following solutions set: 4.1. Impedance Modeling. With the traditional design of the z H zH IDA-PBC controller outlined, the digital control is ignored. d d 􏼨x|x ∈ R , R − J 􏼁 � 0􏼩. (25) d d (e system is modeled with equivalent Laplace transform for zx zx the controller equations to select the appropriate parameters From the LaSalle invariant set theorem [22], the equi- of injected damping gains considering the effect of dis- cretization delay. Based on (20), it is possible to describe the librium point x is considered asymptotically stable. (e energy exchange with the network has been included in the IDA-PBC controller as a three-stage controller for LCL- energy function of the closed-loop system. When the system filtered inverters in the following matrices: ∗ ∗ u i i u E 0 r ωL 1d Cd d 1d 1 1e ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥⎡ ⎢ ⎤ ⎥ ⎢ ⎥⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎡ ⎣ ⎤ ⎦⎢ ⎥ ⎡ ⎣ ⎤ ⎦⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ � − + , (26) ∗ ∗ u i i u 0 E −ωL r q 1q 1e 1 1q Cq ∗ ∗ i F 0 u r u i ωC 1d Cd Cd 2d 2 e ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ �􏼢 􏼣 −􏼢 􏼣􏼢 􏼣+􏼢 􏼣, (27) ∗ ∗ i 0 F u −ωC r u i 1q Cq Cq 2q e 2 ∗ ∗ u i i v G 0 r ωL 2d pccd Cd 2d 3 2e ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ � − + , (28) 􏼢 􏼣 􏼢 􏼣􏼢 􏼣 􏼢 􏼣 ∗ ∗ u i i v 0 G −ωL r 2q pccq Cq 2q 2e 3 where E � R + r , F � r , and G � R + r . (en, the control signal is sent to the LCL filter-based in- 1e 1 2 2e 3 In Figure 2, the delay in the system is added after the verter depicted in Figure 2: deduced reference control law, resulting in the following expression [8]: u u d d ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ � G . (30) sin(ωT/2) d ∗ − 1.5Ts u u G (s) � e , (29) (ωT/2) where T represents the sampling period. (e delay G (s) is (e LCL-filtered inverter system model is expressed as written as G to simplify the expression in the following. follows: d 6 International Transactions on Electrical Energy Systems i i u v 0 ωL A 0 A 0 A 0 2d 2 2d Cd pccd ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ � 􏼢 􏼣􏼢 􏼣 + 􏼢 􏼣 − 􏼢 􏼣 , (31) i −ωL 0 0 A i 0 A u 0 A v 2q 2q Cq pccq u u i i 0 ωC B 0 B 0 B 0 Cd Cd 1d 2d � + − , (32) 􏼢 􏼣 􏼢 􏼣􏼢 􏼣􏼢 􏼣 􏼢 􏼣􏼢 􏼣 􏼢 􏼣􏼢 􏼣 u u i i −ωC 0 0 B 0 B 0 B Cq Cq 1q 2q i i u u 0 ωL D 0 D 0 D 0 1d 1d d Cd 􏼢 􏼣 � 􏼢 􏼣􏼢 􏼣􏼢 􏼣 + 􏼢 􏼣􏼢 􏼣 − 􏼢 􏼣􏼢 􏼣, (33) i −ωL 0 0 D i 0 D u 0 D u 1q 1q q Cq where A � (1/sL + R ), B � (1/sC), and D � (1/sL + R ). Substituting (30) to (33), u and u can be canceling, and d q 2 2 1 1 it can be derived that i u u 1 −ωL D D × G 0 D 0 1d Cd 1 d d ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 􏼢 􏼣 � 􏼢 􏼣 − 􏼢 􏼣 . (34) i u u ωL D 1 0 D × G 0 D 1q q Cq 1 d ∗ ∗ (en, substituting (33) to (26), u and u can be can- d q celing, and it can be obtained that i i 1 −ωL D DE × G 0 1d 1 d 1d ⎡ ⎢ ⎤ ⎥ ⎡ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥⎢ ⎥ ⎡ ⎢ ⎤ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎣ ⎦ ωL D 1 0 DE × G i 1q 1q 1 d (35) i u r D × G ωL D × G D G − 1 􏼁 0 1d Cd 1 d 1e d d ⎢ ⎥⎡ ⎢ ⎤ ⎥ ⎢ ⎥⎡ ⎢ ⎤ ⎥ ⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ − + . i u −ωL D × G r D × G 0 D G − 1􏼁 1q Cq 1e d 1 d d Based on (34), a description of the inner loop response where ψ and φ are the coupling components on the d-q axis would be as follows: and G and Y represent the closed-loop transfer function o o and the inverter output admittance of the system, i i u X ψ Y −φ 1d i 1d i Cd i i respectively. ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ � − , (36) 􏼢 􏼣 􏼢 􏼣 i −ψ X i φ Y u 1q Cq i i 1q i i where ψ and φ are the coupling components d-q axis and i i 4.2. Proposed Parameters’ Design Procedure. As can be seen X and Y correspond to the transfer function of closed- i i from Figure 2, the IDA-PBC controller offers a three-loop loop and output admittance of the inner loop, structure for the LCL inverter system that incorporates respectively: three gains (r , r , r ) that must be determined. Traditional 1 2 3 IDA-PBCs can achieve stable equilibrium conditions with DE × G 1 + r D × G energy shaping techniques. However, the design of inter- d 1 d X � , (37) 2 2 connected damping gains is not guided by any specific design 1 + r D × G 􏼁 + ωL D − ωL D × G 􏼁 1 d 1 1e d procedure. (e following is a step-by-step design procedure for IDA-PBC gains that aims for fast dynamic response and −D G − 1 􏼁 1 + r D × G 􏼁 d 1 d Y � . (38) i high level of robustness against external disturbances. 2 2 1 + r D × G 􏼁 + ωL D − ωL D × G 􏼁 1 d 1 1e d Similarly, combining functions (26)-(33), the equivalent 4.2.1. Design of the Inner Loop Gain r . As the gain of the response of the overall controller can be expressed in (39). inner loop, r should be designed to achieve a fast response. (e derivation process is omitted here: For convenience, a first-order inertial model is substituted for the delay in the design of inner loop. (us, the closed- i i v G ψ Y −φ 2d pccd 2d o o ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ loop transfer function of the inner loop is expressed in � 􏼢 􏼣 − 􏼢 􏼣 , (39) i i v −ψ G φ Y 2q o 2q o pccq function (40): International Transactions on Electrical Energy Systems 7 Re (Yo) Re (Yo) 0.4 0.05 0.2 0.2 0.4 (a) (b) Re (Yo) Re (Yo) 0. 1 0.05 0.05 0.00 0.0 0.2 3 10000 0.4 (c) (d) Figure 3: Real part of LCL-filtered inverter output admittance with IDA-PBC when (a) r � 25 and r varies, (b) r � 0.15 and r varies, 3 2 2 3 (c) r �10 and r varies, and (d) r � 0.1 and r varies. 3 2 2 3 i R + r inverter system impedance with control delay to ensure a 1 1 1 � , (40) stable system. Based on passivity theory, within switching i (1.5Ts + 1) sL + R 􏼁 + r 1 1 1 frequency, if positive real part of system output impedance can be ensured, the system is passive [5]. For the analysis, r ≈ , ξ � 1. (41) 2 mathematics software can be applied to calculate closed-loop 6ξ T transfer function G (s) and output admittance Y (s) for the o o outer loop. In this process, the delay function is substituted It can be seen from (40) that the inner loop is a second- by Euler’s formula. During the steady state condition, i / order system. According to classical control theory, the 2d i � 1 and i /i � 0. Based on function (39), ignoring the critically damped case is preferred to reach shortest time for 2d 2d 2q coupling items, G (s) is expressed as follows: the system. Since the critical damping ratio is equal to 1, then After deciding on r � 2, the relationship between fre- r is chosen as 2. It is should, however, be noted that PI 1 quency, the real part of the inverter output admittance, and regulator can be applied instead of the damping r with r or r can be plotted. (e value of r is not independent, but proper integral coefficient to remove the steady-state error. 2 3 2 varies with value of r . Assume that r is ensured and then 3 3 determine the approximate range of r . Likewise, select the 4.2.2. Design of the Middle Loop Gain r and the Outer Loop middle value of r within the stable range, and determine the 2 2 Gain r . Here, design of r and r relies on passivity of the stable range of r again. In addition, the system closed-loop 3 2 3 1 f (Hz) (Hz) f (Hz) f (Hz) 8 International Transactions on Electrical Energy Systems Pole-Zero Map Pole-Zero Map 1 1 0.8 0.8 0.6 r =0.15 0.6 increase r from r =25 0.05 to 0.25 0.4 0.4 increase r from 0.2 0.2 10 to 40 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Real Axis Real Axis (a) (b) Pole-Zero Map Pole-Zero Map 1 1 0.8 0.8 0.6 0.6 r =0.1 r =10 0.4 3 0.4 increase r from increase r from 2 3 0.2 0.2 0.1 to 0.8 5 to 50 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Real Axis Real Axis (c) (d) Figure 4: Closed-loop zero-pole maps of LCL-filtered inverter with IDA-PBC when (a) r � 25 and r varies, (b) r � 0.15 and r varies, 3 2 2 3 (c) r �10 and r varies, and (d) r � 0.1 and r varies. 3 2 2 3 poles are drawn with different r and r . (is is demonstrated be reached by reverse-seeking, assuming that r is 0.1 and r 1 2 2 1 in Figures 3 and 4, where two groups of r and r are selected is 2, as shown in Figure 3(d). After the iterations, r and r 1 2 2 3 for verification. are selected as 0.15 and 15, consequently. (e overall flow diagram of the proposed design procedure for LCL-filtered i XBFG 2d grid-connected inverter with IDA-PBC is shown in Figure 5. � . (42) −1 1 − BY + Br X􏼁A + XBFr + B − XB􏼁 According to the figures of closed-loop pole-zero maps 2d 2 3 and output admittance real parts, the calculated ranges of parameters based on the internal stability of the system (e relationship between r , inverter output admittance, basically coincide with the ranges calculated according to and frequency is illustrated in Figure 3(a), assuming that r is the passivity of the output admittance. Due to the fact that 25. In order to ensure positive real part of Y (s), r should be the energy function of the closed-loop system represents o 2 less than 0.19, which can also be proved with the closed-loop the physical state of energy at the equilibrium point and zero-pole maps shown in Figure 4(a). accounts for external energy input in its entirety, with the (en, assuming that r is 0.15, reverse-seek the range of selected parameters, the real part of the output admittance r , as demonstrated in Figures 3(b) and 4(b). In the second can be always positive and passive within switching fre- round, r is selected as 10, and the stable range of r are quency. Hence, the overall system can remain stable re- 3 2 illustrated in Figures 3(c) and 4(c). (en, the range of r can gardless of the variation of the grid impedance. Imaginary Axis Imaginary Axis Imaginary Axis Imaginary Axis International Transactions on Electrical Energy Systems 9 Calculate inner loop transfer function Select r Calculate r according to the optimal damping ratio Calculate the admittance model of the whole system Closed-loop admittance modeling Calculate the real part Verify internal stability Determine r and r of output admittance 2 3 Initialize r to obtain the range of r to 3 2 meet the passivity requirements Select the average value of r and use 2 Verify the range of r with passivity to determine the range of r closed-loop pole-zero maps Select the average value of r and use Verify the range of r with passivity to determine the range of r closed-loop pole-zero maps No Iterations times and requirements reached? Yes Get the final r , r , r 1 2 3 Figure 5: Flow diagram of the proposed design procedure for LCL-filtered grid-connected inverter with IDA-PBC. Table 2: Controller parameters. Symbols Components Value r Inner loop damping 2 r Middle loop damping 0.15 r Loop damping 15 5i v 2a ga -50 -100 -150 L =3.6 mH, C =0 uF L =3.6 mH, C =4 uF L =0 mH, C =0 uF g g g g g g 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Time (s) Figure 6: Simulated results when grid impedance varies with proposed IDA-PBC strategy. 10 International Transactions on Electrical Energy Systems ga 5i 2a -50 -100 -150 L =0 mH, C =0 uF L =3.6 mH, C =0 uF L =3.6 mH, C =4 uF g g g g g g 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Time (s) Figure 7: Simulated results when grid impedance varies with traditional IDA-PBC strategy. 15 15 10 10 5 5 0 0 -5 -5 -10 -10 -15 -15 0 0.01 0.02 0.03 0 0.01 0.02 0.03 Time (s) Time (s) (a) (b) -5 -10 -15 0 0.01 0.02 0.03 Time (s) (c) Figure 8: Simulated results of transient responses of grid currents under balanced grid when (a) L � 0 mH and C � 0 uF, (b) L � 4.8 mH g g g and C � 0 uF, and (c) L � 4.8 mH and C � 3 uF. g g g impedance is zero for 0< t<0.06s, and L is set as 3.6mH for 5. Simulation Results 0.06s< t<0.12s, and then, L � 3.6mH and C � 4uF for g g Simulations were conducted with selected system parameter 0.12s< t<0.18s. (e inductor and the capacitor are added in values from Table 1 and controller parameters from Table 2 in the grid at 0.6s and 0.12s. (e system returns to the normal order to evaluate the effectiveness of the proposed strategy. state after one period. (e injected currents are stable and clearly sinusoidal in three periods. Figure 7 depicts the results of the system when grid impedance varies with traditional IDA- 5.1. Under Balanced Grid Voltage. Figure 6 depicts the PBC strategy. It can be seen that traditional IDA-PBC strategy waveforms results of system when implementing the pro- can be stable in strong grid and inductive grid, but it is posed IDA-PBC strategy under the steady state. (e grid unstable under some capacitive grid impedance cases. (e Current (A) Current (A) Current (A) International Transactions on Electrical Energy Systems 11 i [20A/div] v [100V/div] 2a ga -100 -200 L =4.8 mH, C =0 uF L =4.8 mH, C =3 uF L =0 mH, C =0 uF g g g g g g 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Time (s) Figure 9: Simulated results of grid voltages and grid currents when grid impedance varies under unbalanced grid. -10 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s) Figure 10: Simulated results of grid currents when C � 7 uF; L varies from 0 mH to 10mH. g g -10 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s) Figure 11: Simulated results of grid currents when L � 3 mH; C varies from 0 uF to 15 uF. g g proposed IDA-PBC strategy is more robust under com- v [50 V/div] g i [10 A/div] gb plex grid impedance condition. ga (e performance of the proposed method during a tran- 2a sient response is also tested under strong grid, inductive grid, and complex grid, as illustrated in Figure 8. In this analysis, the peak value of the reference current is reduced from 12.86 A to 6.28 A at 0.01s. In the presence of a strong grid, the currents can track the reference value immediately, and even when the grid is weak or capacitive, the response time is also less than 5 ms. 5.2. Under Unbalanced Grid Voltage. Caused by the grid faults or sudden load changes, the voltage dips are probably happen and then bring challenges to the control of the grid. Time [5 ms/div] Figure 9 shows the results of injected currents when v and ga Figure 12: Experimental results of grid voltages and currents with v drop by 36% and 18%, with v � 70(RMS) and gb ga L � 3.6 mH and C � 0 uF. g g v � 90(RMS), respectively. It is apparent that grid currents gb are still sinusoidal and balanced with the tracked reference value under unbalanced grid. L varies from 0 to 10mH within 0.1s. Figure 11 depicts the corresponding results of the grid currents when L is set as 3mH 5.3. Grid Impedance Variation in a Wide Range. and C varies from 0 to 15uF within 0.1s. It is clear that the Figure 10 illustrates a simulation of grid currents to demonstrate system can inject high quality and stable currents both in in- the robustness of the proposed control algorithm under a wide ductive and capacitive grid in a wide range with the selected range of grid impedance variations when C is fixed as 7uF and parameters for the IDA-PBC method. Grid current (A) Grid current (A) 12 International Transactions on Electrical Energy Systems v [50 V/div] i [10 A/div] g 2 gb ga 2a v v gb ga 2a Time [5 ms/div] v [50 V/div] Time [5 ms/div] i [10 A/div] Figure 13: Experimental results of grid voltages and currents with Figure 16: Experimental results of transient responses of grid � 3.6 mH and C � 6 uF. g g currents with L � 3.6 mH and C � 6 uF. g g experimental setup of 3-KW/110-V/three-phase grid in- verter was built to test. (e dc-link source to the Danfoss FC302 inverter is provided by Chroma dc power supply. (e power grid is simulated by a programmable ac source, and a dSPACE DS1007 platform is utilized to implement the control algorithm. Except the system parameters listed in Table 1, the selected parameters of r , r , and r of the IDA- 1 2 3 ga PBC controller are 2, 0.15, and 15, respectively. Based on the v i gb 2a simulation results, it is clear that the system is capable of performing optimally in a balanced grid. (e experimental results of the proposed controller are displayed here under the unbalanced voltage condition. (e voltage of phase A is emulated by a 36% dips with v � 70V (RMS). ga A study of experimental grid currents and voltages v [50 V/div] i [10 A/div] Time [5 ms/div] g 2 under the steady state is presented in Figures 12 and 13. (e grid currents can be controlled to be nearly sinusoidal Figure 14: Experimental results of transient responses of grid with the proposed method in both inductive and capac- currents with L � 0 mH and C � 0 uF. g g itive grids, even though the grid voltages are unbalanced and distorted. To investigate the performance of transient responses with proposed IDA-PBC and selected control parameters, the experimental results of grid currents when the reference current increases under difference grid impedances are demonstrated in Figure 14, 15, and 16. In all cases, the injected grid currents are stable and sinusoidal. (e grid current can rapidly track the reference without gb ga 2a the settling time under the ideal strong grid. Even with the inductive and capacitive unbalanced grid, the injected currents can still track the reference current and the re- sponse time is less than 3 ms. 7. Conclusion v [50 V/div] i [10 A/div] g Time [5 ms/div] A modified IDA-PBC strategy and design method for cal- Figure 15: Experimental results of transient responses of grid culating damping gains based on an impedance model are currents with L � 3.6 mH and C � 0 uF. g g presented in this study. Simulations and experimental tests have been used to determine the effectiveness of the pro- posed controller, from which the following conclusion can 6. Experimental Validation be drawn. To further validate the effectiveness of the IDA-PBC for (1) (e stability of IDA-PBC with the assignment of the LCL-filtered inverter with selected parameters, an desired energy function to the closed-loop system is International Transactions on Electrical Energy Systems 13 coinciding to the passivity of the closed-loop system r : (e middle loop gain output admittance. It is a good feature to allow this r : (e outer loop gain. control method to be widely used in weak grid. (e optional range of injected damping parameters can Data Availability be selected based on the principle to satisfy the output impedance is passive. (e authors confirm that the data supporting the findings of (2) (e design procedure of IDA-PBC considers the this study are available within the article. effect of control delay and a more flexible inter- connection matrix design method is proposed. Conflicts of Interest With the proposed IDA-PBC strategy, the inverter (e authors have no conflicts of interest to declare. system is passive within the switching frequency, which allows high quality current to be injected to Acknowledgments the grid, regardless of large variations in grid impedance and unbalanced grid voltages. 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Stabilization Design of Three-Phase LCL-Filtered Grid-Connected Inverter Using IDA-PBC Controller

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Copyright © 2022 Min Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi International Transactions on Electrical Energy Systems Volume 2022, Article ID 7515321, 14 pages https://doi.org/10.1155/2022/7515321 Research Article StabilizationDesignofThree-PhaseLCL-FilteredGrid-Connected Inverter Using IDA-PBC Controller 1 1 1 1 1 Min Huang , Zhicheng Zhang , Fan Chen , Weimin Wu , Zhilei Yao , and Frede Blaabjerg Department of Electrical Engineering, Shanghai Maritime University, Shanghai 201306, China Department of Energy Technology, Aalborg University, Aalborg 9220, Denmark Correspondence should be addressed to Min Huang; minhuang@shmtu.edu.cn Received 7 December 2021; Revised 24 February 2022; Accepted 15 March 2022; Published 15 April 2022 Academic Editor: Tomislav Capuder Copyright © 2022 Min Huang et al. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. With the aim of improving the stability of renewable energy system with high permeability in the weak grid, a modified passivity- based control based on interconnection and damping assignment (IDA) is presented for LCL-filtered grid-connected inverters to eliminate the interactive resonances. (e object is modeled in the form of port-controller Hamiltonian, including the partial differential of stored energy function. On the premise of ensuring Lyapunov stability, a more flexible interconnection matrix design method is applied to simplify the design process. (e closed-loop stability is ensured with the selected Hamiltonian energy function at the desired balanced point. Moreover, a step-by-step design process for damping gains is provided to guarantee stable operation and fast dynamic response under variation of complex grid impedance. With the designed injected damping pa- rameters, considering the effect of delay, it is possible for the real part of the inverter output admittance to be positive within the switching frequency. (e performance and robustness of the proposed method for LCL-filtered inverter system are validated via simulated and experimental results under both unbalanced and balanced grid conditions. stability with the grid [5]. During actual operation, grid 1. Introduction impedance is varying, resulting in stimulated resonances for In renewable energy generation system, the grid-connected the stable inverter system during independent operation and inverter has become one of the core parts to interface grid even undesirable tripping. Recently, the concept of passivity [1]. Its output current harmonic distortion is an important theory was applied to access the stability of grid-connected technical indicator; thus, LCL filter with low-pass charac- inverters in frequency domain, and many research works teristics is usually connected to inject high quality current. have been proposed to extend nonnegative real part of in- LCL filter has better filtering effect, smaller system volume, verter output admittance [5–10]. It indicates that the res- and lower loss at the same cost compared with L filter [2]. onance will not be stimulated if the interactive point falls in the passive region. Most control methods mainly consider However, as a result of inherent resonance of high-order filter, the circuit is prone to oscillating or even unstable passive regions up to the Nyquist frequency. However, outputs. [3] (erefore, extra requirements are put forward passivity analysis and instability of inverter admittance for the controller design for LCL-filtered inverter system. above the Nyquist frequency should also be considered [8]. Additionally, 1-beat or 1.5-beat delay could be brought by Currently, most closed-loop control strategies rely on the discrete control and pulse width modulation [4]. (e control classic linear control theory, which can be used to accurately delay will typically reduce the phase margin as well as design the controller parameters and evaluate the influence complicate the design of the controller. of the delay. (ese control methods are however difficult to Apart from the stability of the inverter system itself, fully eliminate interactive resonances under complex weak another concern of the controller design is the interactive grid, and the traditional proportional integral control does 2 International Transactions on Electrical Energy Systems demonstrated by simulations and experimental results. not have sufficient disturbances rejection capability. Com- pared with the classical linear control theory, the modern Furthermore, the study found that IDA-PBC also performed well in unbalanced grid. Section 7 concludes the study. nonlinear control theory has better robustness and per- formance, such as sliding mode control [11, 12], predictive control [13, 14], and passivity-based control (PBC) [15–26]. 2. System Description and PCH Modeling PBC has proven useful in the design of robust controllers for railway systems [19, 20], renewable generation systems [21], 2.1. Description of the System. Figure 1 depicts an inverter etc. In terms of the PBC design process, energy formation with a power supply voltage U and is filtered by a LCL filter dc and damping injection are considered to be of obvious including inverter-side inductor L , filter capacitor C, and physical significance [17], and two branches are proposed by grid-side inductor L . (e parasitic resistances of L and L 2 1 2 the Euler–Lagrange (EL) model and the port-controller are represented by R and R , respectively. (e voltage v is 1 2 pcc Hamiltonian (PCH) model, called EL-PBC and IDA-PBC, used for the synchronization of the grid current. Addi- respectively. [22–26] (e modeling process of the EL-PBC tionally, the inverter-side current i , the capacitor voltage u , 1 C method is simpler, but IDA-PBC offers more flexibility in and the grid-side current i should also be measured to the type of damping interconnection and the modeling control the injected current as well as suppress the reso- process. Moreover, for the IDA-PBC, fast convergence speed nances. It should be noted that the state observer or Kalman of the error energy function can be determined by calcu- filter can be used to reduce the number of required sensors lating the derivative of the closed-loop energy function and by estimating state variables [9, 11]. (e equivalent grid the injected damping [23]. For EL-PBC, when the error is reactance of L and C is connected to the grid at the point of g g large, the convergence speed is fast, and injected damping common coupling (PCC), which can take a wide range of plays the main role. In contrast, the convergence speed values. (erefore, the controller must be robust enough to becomes slower and the role of injected damping becomes withstand grid disturbance. Table 1 summarizes basic system weak when the error is small. Towards this end, IDA-PBC parameters for analysis. has been adopted in this study so as to achieve better di ⎧ ⎪ 1k performance under complex grid conditions. ⎪ ⎪ L + R i + u � u , 1 1 1k Ck ik dt However, it is often overlooked that time delay can affect ⎪ IDA-PBC [25]. As to the design of damping matrix of IDA- PBC, few research studies analyze the specific design du Ck (1) C + i − i � 0, 2k 1k guideline and the try-and-error method is usually adopted. dt (erefore, for the purpose of assessing the stability of weak grid, the detailed design of digital controlled IDA-PBC is di 2k L + R i − u � −v . necessary. A brief summary of the main contributions is as 2 2 2k Ck pcck dt follows. In accordance with Figure 1, the equations for a three- (1) On the premise of ensuring Lyapunov stability, to phase LCL filtered inverter system can be derived from simplify the design process for PCH model-based Kirchhoff’s law in a stationary abc frame as follows:where PBC, a more flexible interconnection matrix design k stands for the cases in abc coordinates. (rough the method is used. rotation transformation, from (1), the mathematical (2) (is study demonstrates how to select the injected model of the grid-connected inverter in d-q coordinates damping considering the effect of control delay using appears as frequency-domain passivity theory based on linear di control design. ⎪ 1d L � u − R i + ωL i − u , 1 id 1 1d 1 1q Cd dt (3) With the proposed controller parameters’ design method, the nonnegative real part of the LCL-filtered di ⎪ 1q inverter output admittance can be achieved within L � u − R i − ωL i − u , 1 iq 1 1q 1 1d Cq switching frequency. If the passivity of subsystem is dt ensured, then the stability of the whole interlinked du ⎧ ⎪ Cd system in parallel form is guaranteed regardless of C � i − i + ωCu , 1d 2d Cq dt grid impedance. (2) Moreover, in Section 2, a brief description of the LCL- du ⎪ Cq filtered inverter system is given, followed by the design of C � i − i − ωCu , 1q 2q Cd dt IDA-PBC methodology in Section 3. Instead of the original antisymmetric matrix, the interconnection matrix is sim- di 2d ⎧ ⎪ plified by using the upper triangular matrix with zero corner L � u − R i + ωL i − v , ⎪ 2 Cd 2 2d 2 2q pccd ⎪ dt elements. In order to precisely design the injected damping gains, in Section 4, frequency-domain passivity theory is ⎪ di applied with the derived impedance model of IDA-PBC. In ⎪ 2q L � u − R i − ωL i − v . 2 Cq 2 2q 2 2d pccq Section 5 and Section 6, system performance is dt International Transactions on Electrical Energy Systems 3 Table 1: System parameters. LCL-filter PCC L , R L , R g Symbols Components Value 1 1 2 2 U DC bus voltage 350 V dc 1 C U Grid voltage 110V (RMS) dc f Grid frequency 50 Hz i pcc i u 2 Grid 1 C impedance L Inverter-side inductor 1.2 mH C Filter capacitor 6 uF L Grid-side inductor 1.2 mH SV IDA-PBC PWM f Sampling frequency 10 kHz f Switching frequency 10 kHz sw Figure 1: Structure of LCL-filtered three-phase grid-connected inverter. 3. IDA-PBC Controller Design 2.2. PCH Modeling. In order to construct a robust controller (e primary goal of the controller is to find the Hamiltonian for the above system, a PCH model with dissipation can be energy storage function H (x), interconnection matrix J (x), a a used [24]. It is possible to express the system’s PCH model as and damping matrix R (x), which satisfy the following follows: equation with a balance point at x : zH(x) (3) x _ � [J(x) − R(x)] + g(x)u, zH(x) zx 􏼂J (x) − R (x)􏼃K(x) � −􏼂J (x) − R (x)􏼃 + g(x)u, d d a a zx where J(x) represents the system interconnection prop- (8) erties and normally satisfy formula J(x) � -J(x) , R(x) where K(x) � [zH (x)/zx], J (x) � J(x) + J (x), indicates the system dissipation characteristics, and a d a R (x) � R(x) + R (x), and H (x) is the undetermined H(x) is the Hamiltonian function; u indicates the ex- d a function. change of energy between the outside world and the Moreover, in order to ensure x is locally stable equi- system. (e coefficient matrix g(x) stands for the inter- librium point, the following conditions should also be sat- connections. (e following is a definition of system isfied [15]. variables: (1) In traditional design, strict structure preservation is x � 􏼂 x x x x x x 􏼃 1 2 3 4 5 6 provided (4) L i L i Cu Cu L i L i � 􏽨 􏽩 . 1 1d 1 1q Cd Cq 2 2d 2 2q T ⎧ ⎨ J (x) � J(x) + J (x) � −􏼂J(x) + J (x)􏼃 , d a a (9) (e following is the expression for each of the R (x) � R(x) + R (x) � 􏼂R(x) + R (x)􏼃 . d a a vectors and matrices for the LCL-filter inverter according to (2): (2) K(x) is the integrated gradient of a scalar function: 0 ωL −1 0 0 0 1 zK(x) zK(x) ⎡ ⎢ ⎤ ⎥ (10) ⎢ ⎥ � 􏼠 􏼡 . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ zx zx ⎢ −ωL 0 0 −1 0 0 ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 0 0 ωC −1 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (3) Expect the balance x ; the desired dynamics is ⎢ ⎥ J � ⎢ ⎥, ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 1 −ωC 0 0 −1 ⎥ ⎢ ⎥ achieved if K(x) is satisfied: ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 1 0 0 ωL ⎥ ⎢ ⎥ zH(x) ⎢ 2 ⎥ ⎢ ⎥ ∗ ⎣ ⎦ (11) K x 􏼁 � − | ∗ , (5) x�x zx 0 0 0 1 −ωL 0 1 0 0 0 0 0 H (x) � H(x) + H (x). (12) d a ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 1 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ g(x) � ⎢ ⎥ , ⎢ ⎥ (4) At the balance point, K(x) should satisfy the fol- ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 −1 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ lowing function based on the Lyapunov stability: 0 0 0 0 0 −1 zK(x) z H(x) u � 􏽨u , u , v , v 􏽩 , (13) d q pccd pccq | > − | . ∗ ∗ x�x x�x (6) 2 zx zx R � diag􏼂 R R 0 0 R R 􏼃. 1 1 2 2 For EL-PBC, J(x) is fixed and the damping is injected (e stored energy in the LCL filter can be expressed as with constructed energy function. For IDA-PBC, it is more flexible to design J (x) and R (x) corresponding to desired 2 2 2 2 2 2 a a x + x x + x x + x 1 2 3 4 5 6 energy function. At the stable equilibrium point x , the (7) H(x) � + + . 2L 2C 2L 1 2 closed-loop system based on PCH is described as [22] 4 International Transactions on Electrical Energy Systems and x can gradually reach the equilibrium points, which are zH (x) 4 x _ � J (x) − R (x) . (14) 􏼂 􏼃 d d defined as follows: zx ∗ ∗ ∗ ∗ ∗ ∗ (17) At the desired x , the tracking error of selected Ham- x∗ � 􏽨L i , L i , Cu , Cu , L i , L i 􏽩 . 1 1d 1 1d Cd Cq 2 2d 2 2q iltonian function H (x) should be minimum. (e derivation (e derivative of H (x) can be defined as K(s) and of the closed-loop energy function H (x) is depicted as d corresponding components of K(s) can be viewed as follows: zH (x) _ T H (x) � 􏼢 􏼣 x _ (18) d K(x) � 􏼂k (x), k (x), k (x), k (x), k (x), k (x)􏼃 , 1 2 3 4 5 6 zx (15) ∗ ∗ x x T ⎧ ⎪ 1 2 k (x) � − , k (x) � − , zH (x) zH (x) ⎪ 1 2 d d � − R (x) − J (x) . ⎪ L L 􏼢 􏼣 􏼂 􏼃 1 1 d d ⎪ zx zx ∗ ∗ x x 3 4 (19) Based on Lyapunov’s second criterion, the derivative of k (x) � − , k (x) � − , ⎪ 3 4 C C H (x) of the closed-loop energy function must be negative to ∗ ∗ attain an asymptotically stable equilibrium point. Conven- ⎪ x x 5 6 ⎩ (x) � − , k (x) � − , 5 6 tional, J is designed as positive definite symmetric matrix L L 2 2 based on structure preservation in (9). A more flexible method is proposed to design J in this study. (e derivative When J , R , and K(x) are substituted to (1), reference d a a of the energy function can be less than zero if [R -J ] is a values for the state variables of the system can be found in d d positive definite matrix. So, if injected damping R is pos- (20) which determined the passive controller u. As a result of itive, J can be designed as a lower triangular matrix or an (20), as shown in Figure 2, the cascaded three loops con- upper triangular matrix with zero diagonal elements to troller can be depicted. It is worth mention that “L ,” “L ,” 1e 2e simplify the design procedure, and it is also capable of “C ,” “R ,” and “R ” are the initial filter parameters given in e 1e 2e ensuring that H (x)< 0. So, J , J , and R are defined as the controller, and the subscript “e” is used to differentiate a d a with actual parameters of thr physical filter. 0 −ωL 1 0 0 0 ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ ∗ ∗ ⎢ ⎥ ⎢ ωL 0 0 1 0 0 ⎥ ⎢ ⎥ u � u − ωL i + r i − i 􏼁 + R i , ⎢ ⎥ ⎢ 1 ⎥ ⎧ ⎪ cd 1e 1q 1 1d 1e ⎢ ⎥ d 1d 1d ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ∗ ∗ ∗ ⎢ ⎥ ⎢ 0 0 0 −ωC 1 0 ⎥ ⎪ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ u � u + ωL i + r 􏼐i − i 􏼑 + R i , ⎢ ⎥ ⎢ ⎥ q cq 1e 1d 1 1q 1q 1e 1q ⎢ ⎥ ⎪ ⎢ ⎥ J � ⎥, ⎢ ⎥ ⎪ a ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ 0 0 ωC 0 0 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ ∗ 2d ∗ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ i � − ωC u + r u − u , ⎢ ⎥ 􏼁 ⎢ ⎥ ⎨ ⎢ ⎥ 1d i e Cq 2 Cd Cd ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (20) ⎢ 0 0 0 0 0 −ωL ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎪ ⎣ ⎦ 2q ∗ ∗ i � + ωC u + r 􏼐u − u 􏼑, ⎪ 1q i e Cd 2 Cq Cq 0 0 0 0 ωL 0 ∗ ∗ u � v − ωL i + r i − i 􏼁 + R i , Cd pccd 2e 2q 3 2d 2d 2e 2d 0 0 0 0 0 0 ⎪ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎩ ⎢ ⎥ ∗ ∗ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ u � v + ωL i + r i − i + R i . ⎢ ⎥ 􏼐 􏼑 ⎢ ⎥ ⎢ 0 0 0 0 0 0 ⎥ Cq pccq 2e 2d 3 2q 2q 2e 2q ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ By using (11) and (12), it is possible to calculate the ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ J ⎢ ⎥, R � ⎢ ⎥ � diag􏼂 r r r r r r 􏼃. ⎢ ⎥ d ⎢ ⎥ a 1 1 2 2 3 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ amount of energy injected into the system by the controller: ⎢ ⎥ ⎢ ⎥ ⎢ 0 1 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎥ ⎢ 0 0 1 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x x x x x x ⎣ ⎦ 1 2 3 4 5 6 H (x) � − x − x − x − x − x − x . a 1 2 3 4 5 6 0 0 0 1 0 0 L L C C L L 1 1 2 2 (16) (21) (e elements of R should be positive. (rough the Accordingly, the closed-loop Hamiltonian energy ∗ ∗ control states x ⟶x and x ⟶x , the states x , x , x , function for this system is as follows: 5 5 6 6 1 2 3 ∗ ∗ ∗ ∗ ∗ ∗ 2 2 2 2 2 2 x x x x x x x + x x + x x + x 1 2 3 4 5 6 1 2 3 4 5 6 (22) H (x) � − x − x − x − x − x − x + + + . d 1 2 3 4 5 6 L L C C L L 2L 2C 2L 1 1 2 2 1 2 1 1 1 1 1 1 z H (x) (23) | ∗ � diag􏼢 􏼣> 0. At the balanced point x , zH (x)/zx is equal to 0, and x�x L L C C L L zx 1 1 2 2 the Hessian matrix of H (x) shown in (23) meets the re- quirement of (13): Hence, H (x) has the minimum value at x . Furthermore, d International Transactions on Electrical Energy Systems 5 LCL-filter Controller u u i v pccd i Cd Cd 2d pccd 2d 2e * 1e i + u u 1d + d d i * - -- 2d u * i Cd Cd 2d + + i 1 1 1 1d r r r G (s) 3 2 1 d sL +R sC sL +R 1 1 - - - 2 2 - - - + + Cd 2d ωL ωL ωC 1d ωL ωL 2e 1e ωC e 1 2 Cq i ωL 1q ωL ωL ωL 2q ωC ωC 2e 1e 1 2 e * u u q q + + * - -- i - - - 2q 1 1 1 r r r G (s) 3 2 1 d * + sL +R + * sC sL +R + 1 1 u i 2 2 u Cq 1q Cq + + - -- i + i 2q 1q R R 2e 1e u v v i Cq pccq Cq i pccq 2q 2q Figure 2: Proposed IDA-PBC strategies for LCL-filtered grid-connected inverter. is dissipative, this also implies that it is passive. In order to zH (x) H(x) � x _ 􏼢 􏼣 select the values of the injected damping coefficients, Section zx 4 will analyze the passivity of the closed-loop output ad- (24) mittance using the impedance model. zH (x) zH (x) d d � −􏼢 􏼣 􏼂R (x) − J (x)􏼃 ≤ 0. d d zx zx 4. Impedance Model and Stability Oriented Design And apart from x , there are no solutions of x (t) which can stay in the following solutions set: 4.1. Impedance Modeling. With the traditional design of the z H zH IDA-PBC controller outlined, the digital control is ignored. d d 􏼨x|x ∈ R , R − J 􏼁 � 0􏼩. (25) d d (e system is modeled with equivalent Laplace transform for zx zx the controller equations to select the appropriate parameters From the LaSalle invariant set theorem [22], the equi- of injected damping gains considering the effect of dis- cretization delay. Based on (20), it is possible to describe the librium point x is considered asymptotically stable. (e energy exchange with the network has been included in the IDA-PBC controller as a three-stage controller for LCL- energy function of the closed-loop system. When the system filtered inverters in the following matrices: ∗ ∗ u i i u E 0 r ωL 1d Cd d 1d 1 1e ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥⎡ ⎢ ⎤ ⎥ ⎢ ⎥⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎡ ⎣ ⎤ ⎦⎢ ⎥ ⎡ ⎣ ⎤ ⎦⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ � − + , (26) ∗ ∗ u i i u 0 E −ωL r q 1q 1e 1 1q Cq ∗ ∗ i F 0 u r u i ωC 1d Cd Cd 2d 2 e ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ �􏼢 􏼣 −􏼢 􏼣􏼢 􏼣+􏼢 􏼣, (27) ∗ ∗ i 0 F u −ωC r u i 1q Cq Cq 2q e 2 ∗ ∗ u i i v G 0 r ωL 2d pccd Cd 2d 3 2e ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ � − + , (28) 􏼢 􏼣 􏼢 􏼣􏼢 􏼣 􏼢 􏼣 ∗ ∗ u i i v 0 G −ωL r 2q pccq Cq 2q 2e 3 where E � R + r , F � r , and G � R + r . (en, the control signal is sent to the LCL filter-based in- 1e 1 2 2e 3 In Figure 2, the delay in the system is added after the verter depicted in Figure 2: deduced reference control law, resulting in the following expression [8]: u u d d ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ � G . (30) sin(ωT/2) d ∗ − 1.5Ts u u G (s) � e , (29) (ωT/2) where T represents the sampling period. (e delay G (s) is (e LCL-filtered inverter system model is expressed as written as G to simplify the expression in the following. follows: d 6 International Transactions on Electrical Energy Systems i i u v 0 ωL A 0 A 0 A 0 2d 2 2d Cd pccd ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ � 􏼢 􏼣􏼢 􏼣 + 􏼢 􏼣 − 􏼢 􏼣 , (31) i −ωL 0 0 A i 0 A u 0 A v 2q 2q Cq pccq u u i i 0 ωC B 0 B 0 B 0 Cd Cd 1d 2d � + − , (32) 􏼢 􏼣 􏼢 􏼣􏼢 􏼣􏼢 􏼣 􏼢 􏼣􏼢 􏼣 􏼢 􏼣􏼢 􏼣 u u i i −ωC 0 0 B 0 B 0 B Cq Cq 1q 2q i i u u 0 ωL D 0 D 0 D 0 1d 1d d Cd 􏼢 􏼣 � 􏼢 􏼣􏼢 􏼣􏼢 􏼣 + 􏼢 􏼣􏼢 􏼣 − 􏼢 􏼣􏼢 􏼣, (33) i −ωL 0 0 D i 0 D u 0 D u 1q 1q q Cq where A � (1/sL + R ), B � (1/sC), and D � (1/sL + R ). Substituting (30) to (33), u and u can be canceling, and d q 2 2 1 1 it can be derived that i u u 1 −ωL D D × G 0 D 0 1d Cd 1 d d ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 􏼢 􏼣 � 􏼢 􏼣 − 􏼢 􏼣 . (34) i u u ωL D 1 0 D × G 0 D 1q q Cq 1 d ∗ ∗ (en, substituting (33) to (26), u and u can be can- d q celing, and it can be obtained that i i 1 −ωL D DE × G 0 1d 1 d 1d ⎡ ⎢ ⎤ ⎥ ⎡ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥⎢ ⎥ ⎡ ⎢ ⎤ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎣ ⎦ ωL D 1 0 DE × G i 1q 1q 1 d (35) i u r D × G ωL D × G D G − 1 􏼁 0 1d Cd 1 d 1e d d ⎢ ⎥⎡ ⎢ ⎤ ⎥ ⎢ ⎥⎡ ⎢ ⎤ ⎥ ⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ − + . i u −ωL D × G r D × G 0 D G − 1􏼁 1q Cq 1e d 1 d d Based on (34), a description of the inner loop response where ψ and φ are the coupling components on the d-q axis would be as follows: and G and Y represent the closed-loop transfer function o o and the inverter output admittance of the system, i i u X ψ Y −φ 1d i 1d i Cd i i respectively. ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ � − , (36) 􏼢 􏼣 􏼢 􏼣 i −ψ X i φ Y u 1q Cq i i 1q i i where ψ and φ are the coupling components d-q axis and i i 4.2. Proposed Parameters’ Design Procedure. As can be seen X and Y correspond to the transfer function of closed- i i from Figure 2, the IDA-PBC controller offers a three-loop loop and output admittance of the inner loop, structure for the LCL inverter system that incorporates respectively: three gains (r , r , r ) that must be determined. Traditional 1 2 3 IDA-PBCs can achieve stable equilibrium conditions with DE × G 1 + r D × G energy shaping techniques. However, the design of inter- d 1 d X � , (37) 2 2 connected damping gains is not guided by any specific design 1 + r D × G 􏼁 + ωL D − ωL D × G 􏼁 1 d 1 1e d procedure. (e following is a step-by-step design procedure for IDA-PBC gains that aims for fast dynamic response and −D G − 1 􏼁 1 + r D × G 􏼁 d 1 d Y � . (38) i high level of robustness against external disturbances. 2 2 1 + r D × G 􏼁 + ωL D − ωL D × G 􏼁 1 d 1 1e d Similarly, combining functions (26)-(33), the equivalent 4.2.1. Design of the Inner Loop Gain r . As the gain of the response of the overall controller can be expressed in (39). inner loop, r should be designed to achieve a fast response. (e derivation process is omitted here: For convenience, a first-order inertial model is substituted for the delay in the design of inner loop. (us, the closed- i i v G ψ Y −φ 2d pccd 2d o o ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ loop transfer function of the inner loop is expressed in � 􏼢 􏼣 − 􏼢 􏼣 , (39) i i v −ψ G φ Y 2q o 2q o pccq function (40): International Transactions on Electrical Energy Systems 7 Re (Yo) Re (Yo) 0.4 0.05 0.2 0.2 0.4 (a) (b) Re (Yo) Re (Yo) 0. 1 0.05 0.05 0.00 0.0 0.2 3 10000 0.4 (c) (d) Figure 3: Real part of LCL-filtered inverter output admittance with IDA-PBC when (a) r � 25 and r varies, (b) r � 0.15 and r varies, 3 2 2 3 (c) r �10 and r varies, and (d) r � 0.1 and r varies. 3 2 2 3 i R + r inverter system impedance with control delay to ensure a 1 1 1 � , (40) stable system. Based on passivity theory, within switching i (1.5Ts + 1) sL + R 􏼁 + r 1 1 1 frequency, if positive real part of system output impedance can be ensured, the system is passive [5]. For the analysis, r ≈ , ξ � 1. (41) 2 mathematics software can be applied to calculate closed-loop 6ξ T transfer function G (s) and output admittance Y (s) for the o o outer loop. In this process, the delay function is substituted It can be seen from (40) that the inner loop is a second- by Euler’s formula. During the steady state condition, i / order system. According to classical control theory, the 2d i � 1 and i /i � 0. Based on function (39), ignoring the critically damped case is preferred to reach shortest time for 2d 2d 2q coupling items, G (s) is expressed as follows: the system. Since the critical damping ratio is equal to 1, then After deciding on r � 2, the relationship between fre- r is chosen as 2. It is should, however, be noted that PI 1 quency, the real part of the inverter output admittance, and regulator can be applied instead of the damping r with r or r can be plotted. (e value of r is not independent, but proper integral coefficient to remove the steady-state error. 2 3 2 varies with value of r . Assume that r is ensured and then 3 3 determine the approximate range of r . Likewise, select the 4.2.2. Design of the Middle Loop Gain r and the Outer Loop middle value of r within the stable range, and determine the 2 2 Gain r . Here, design of r and r relies on passivity of the stable range of r again. In addition, the system closed-loop 3 2 3 1 f (Hz) (Hz) f (Hz) f (Hz) 8 International Transactions on Electrical Energy Systems Pole-Zero Map Pole-Zero Map 1 1 0.8 0.8 0.6 r =0.15 0.6 increase r from r =25 0.05 to 0.25 0.4 0.4 increase r from 0.2 0.2 10 to 40 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Real Axis Real Axis (a) (b) Pole-Zero Map Pole-Zero Map 1 1 0.8 0.8 0.6 0.6 r =0.1 r =10 0.4 3 0.4 increase r from increase r from 2 3 0.2 0.2 0.1 to 0.8 5 to 50 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Real Axis Real Axis (c) (d) Figure 4: Closed-loop zero-pole maps of LCL-filtered inverter with IDA-PBC when (a) r � 25 and r varies, (b) r � 0.15 and r varies, 3 2 2 3 (c) r �10 and r varies, and (d) r � 0.1 and r varies. 3 2 2 3 poles are drawn with different r and r . (is is demonstrated be reached by reverse-seeking, assuming that r is 0.1 and r 1 2 2 1 in Figures 3 and 4, where two groups of r and r are selected is 2, as shown in Figure 3(d). After the iterations, r and r 1 2 2 3 for verification. are selected as 0.15 and 15, consequently. (e overall flow diagram of the proposed design procedure for LCL-filtered i XBFG 2d grid-connected inverter with IDA-PBC is shown in Figure 5. � . (42) −1 1 − BY + Br X􏼁A + XBFr + B − XB􏼁 According to the figures of closed-loop pole-zero maps 2d 2 3 and output admittance real parts, the calculated ranges of parameters based on the internal stability of the system (e relationship between r , inverter output admittance, basically coincide with the ranges calculated according to and frequency is illustrated in Figure 3(a), assuming that r is the passivity of the output admittance. Due to the fact that 25. In order to ensure positive real part of Y (s), r should be the energy function of the closed-loop system represents o 2 less than 0.19, which can also be proved with the closed-loop the physical state of energy at the equilibrium point and zero-pole maps shown in Figure 4(a). accounts for external energy input in its entirety, with the (en, assuming that r is 0.15, reverse-seek the range of selected parameters, the real part of the output admittance r , as demonstrated in Figures 3(b) and 4(b). In the second can be always positive and passive within switching fre- round, r is selected as 10, and the stable range of r are quency. Hence, the overall system can remain stable re- 3 2 illustrated in Figures 3(c) and 4(c). (en, the range of r can gardless of the variation of the grid impedance. Imaginary Axis Imaginary Axis Imaginary Axis Imaginary Axis International Transactions on Electrical Energy Systems 9 Calculate inner loop transfer function Select r Calculate r according to the optimal damping ratio Calculate the admittance model of the whole system Closed-loop admittance modeling Calculate the real part Verify internal stability Determine r and r of output admittance 2 3 Initialize r to obtain the range of r to 3 2 meet the passivity requirements Select the average value of r and use 2 Verify the range of r with passivity to determine the range of r closed-loop pole-zero maps Select the average value of r and use Verify the range of r with passivity to determine the range of r closed-loop pole-zero maps No Iterations times and requirements reached? Yes Get the final r , r , r 1 2 3 Figure 5: Flow diagram of the proposed design procedure for LCL-filtered grid-connected inverter with IDA-PBC. Table 2: Controller parameters. Symbols Components Value r Inner loop damping 2 r Middle loop damping 0.15 r Loop damping 15 5i v 2a ga -50 -100 -150 L =3.6 mH, C =0 uF L =3.6 mH, C =4 uF L =0 mH, C =0 uF g g g g g g 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Time (s) Figure 6: Simulated results when grid impedance varies with proposed IDA-PBC strategy. 10 International Transactions on Electrical Energy Systems ga 5i 2a -50 -100 -150 L =0 mH, C =0 uF L =3.6 mH, C =0 uF L =3.6 mH, C =4 uF g g g g g g 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Time (s) Figure 7: Simulated results when grid impedance varies with traditional IDA-PBC strategy. 15 15 10 10 5 5 0 0 -5 -5 -10 -10 -15 -15 0 0.01 0.02 0.03 0 0.01 0.02 0.03 Time (s) Time (s) (a) (b) -5 -10 -15 0 0.01 0.02 0.03 Time (s) (c) Figure 8: Simulated results of transient responses of grid currents under balanced grid when (a) L � 0 mH and C � 0 uF, (b) L � 4.8 mH g g g and C � 0 uF, and (c) L � 4.8 mH and C � 3 uF. g g g impedance is zero for 0< t<0.06s, and L is set as 3.6mH for 5. Simulation Results 0.06s< t<0.12s, and then, L � 3.6mH and C � 4uF for g g Simulations were conducted with selected system parameter 0.12s< t<0.18s. (e inductor and the capacitor are added in values from Table 1 and controller parameters from Table 2 in the grid at 0.6s and 0.12s. (e system returns to the normal order to evaluate the effectiveness of the proposed strategy. state after one period. (e injected currents are stable and clearly sinusoidal in three periods. Figure 7 depicts the results of the system when grid impedance varies with traditional IDA- 5.1. Under Balanced Grid Voltage. Figure 6 depicts the PBC strategy. It can be seen that traditional IDA-PBC strategy waveforms results of system when implementing the pro- can be stable in strong grid and inductive grid, but it is posed IDA-PBC strategy under the steady state. (e grid unstable under some capacitive grid impedance cases. (e Current (A) Current (A) Current (A) International Transactions on Electrical Energy Systems 11 i [20A/div] v [100V/div] 2a ga -100 -200 L =4.8 mH, C =0 uF L =4.8 mH, C =3 uF L =0 mH, C =0 uF g g g g g g 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Time (s) Figure 9: Simulated results of grid voltages and grid currents when grid impedance varies under unbalanced grid. -10 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s) Figure 10: Simulated results of grid currents when C � 7 uF; L varies from 0 mH to 10mH. g g -10 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s) Figure 11: Simulated results of grid currents when L � 3 mH; C varies from 0 uF to 15 uF. g g proposed IDA-PBC strategy is more robust under com- v [50 V/div] g i [10 A/div] gb plex grid impedance condition. ga (e performance of the proposed method during a tran- 2a sient response is also tested under strong grid, inductive grid, and complex grid, as illustrated in Figure 8. In this analysis, the peak value of the reference current is reduced from 12.86 A to 6.28 A at 0.01s. In the presence of a strong grid, the currents can track the reference value immediately, and even when the grid is weak or capacitive, the response time is also less than 5 ms. 5.2. Under Unbalanced Grid Voltage. Caused by the grid faults or sudden load changes, the voltage dips are probably happen and then bring challenges to the control of the grid. Time [5 ms/div] Figure 9 shows the results of injected currents when v and ga Figure 12: Experimental results of grid voltages and currents with v drop by 36% and 18%, with v � 70(RMS) and gb ga L � 3.6 mH and C � 0 uF. g g v � 90(RMS), respectively. It is apparent that grid currents gb are still sinusoidal and balanced with the tracked reference value under unbalanced grid. L varies from 0 to 10mH within 0.1s. Figure 11 depicts the corresponding results of the grid currents when L is set as 3mH 5.3. Grid Impedance Variation in a Wide Range. and C varies from 0 to 15uF within 0.1s. It is clear that the Figure 10 illustrates a simulation of grid currents to demonstrate system can inject high quality and stable currents both in in- the robustness of the proposed control algorithm under a wide ductive and capacitive grid in a wide range with the selected range of grid impedance variations when C is fixed as 7uF and parameters for the IDA-PBC method. Grid current (A) Grid current (A) 12 International Transactions on Electrical Energy Systems v [50 V/div] i [10 A/div] g 2 gb ga 2a v v gb ga 2a Time [5 ms/div] v [50 V/div] Time [5 ms/div] i [10 A/div] Figure 13: Experimental results of grid voltages and currents with Figure 16: Experimental results of transient responses of grid � 3.6 mH and C � 6 uF. g g currents with L � 3.6 mH and C � 6 uF. g g experimental setup of 3-KW/110-V/three-phase grid in- verter was built to test. (e dc-link source to the Danfoss FC302 inverter is provided by Chroma dc power supply. (e power grid is simulated by a programmable ac source, and a dSPACE DS1007 platform is utilized to implement the control algorithm. Except the system parameters listed in Table 1, the selected parameters of r , r , and r of the IDA- 1 2 3 ga PBC controller are 2, 0.15, and 15, respectively. Based on the v i gb 2a simulation results, it is clear that the system is capable of performing optimally in a balanced grid. (e experimental results of the proposed controller are displayed here under the unbalanced voltage condition. (e voltage of phase A is emulated by a 36% dips with v � 70V (RMS). ga A study of experimental grid currents and voltages v [50 V/div] i [10 A/div] Time [5 ms/div] g 2 under the steady state is presented in Figures 12 and 13. (e grid currents can be controlled to be nearly sinusoidal Figure 14: Experimental results of transient responses of grid with the proposed method in both inductive and capac- currents with L � 0 mH and C � 0 uF. g g itive grids, even though the grid voltages are unbalanced and distorted. To investigate the performance of transient responses with proposed IDA-PBC and selected control parameters, the experimental results of grid currents when the reference current increases under difference grid impedances are demonstrated in Figure 14, 15, and 16. In all cases, the injected grid currents are stable and sinusoidal. (e grid current can rapidly track the reference without gb ga 2a the settling time under the ideal strong grid. Even with the inductive and capacitive unbalanced grid, the injected currents can still track the reference current and the re- sponse time is less than 3 ms. 7. Conclusion v [50 V/div] i [10 A/div] g Time [5 ms/div] A modified IDA-PBC strategy and design method for cal- Figure 15: Experimental results of transient responses of grid culating damping gains based on an impedance model are currents with L � 3.6 mH and C � 0 uF. g g presented in this study. Simulations and experimental tests have been used to determine the effectiveness of the pro- posed controller, from which the following conclusion can 6. Experimental Validation be drawn. To further validate the effectiveness of the IDA-PBC for (1) (e stability of IDA-PBC with the assignment of the LCL-filtered inverter with selected parameters, an desired energy function to the closed-loop system is International Transactions on Electrical Energy Systems 13 coinciding to the passivity of the closed-loop system r : (e middle loop gain output admittance. It is a good feature to allow this r : (e outer loop gain. control method to be widely used in weak grid. (e optional range of injected damping parameters can Data Availability be selected based on the principle to satisfy the output impedance is passive. (e authors confirm that the data supporting the findings of (2) (e design procedure of IDA-PBC considers the this study are available within the article. effect of control delay and a more flexible inter- connection matrix design method is proposed. Conflicts of Interest With the proposed IDA-PBC strategy, the inverter (e authors have no conflicts of interest to declare. system is passive within the switching frequency, which allows high quality current to be injected to Acknowledgments the grid, regardless of large variations in grid impedance and unbalanced grid voltages. 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International Transactions on Electrical Energy Systems – Hindawi Publishing Corporation

Published: Apr 15, 2022

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