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

Authors: Min Huang; Zhicheng Zhang; Fan Chen; Weimin Wu; Zhilei Yao; Frede Blaabjerg
Journal: International Transactions on Electrical Energy Systems
 (2022)

---

## Abstract

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. The 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. The 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 parameters, considering the effect of delay, it is possible for the real part of the inverter output admittance to be positive within the switching frequency. The 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.


---

## Body

## 1. Introduction

In renewable energy generation system, the grid-connected inverter has become one of the core parts to interface grid [1]. Its output current harmonic distortion is an important technical indicator; thus, LCL filter with low-pass characteristics is usually connected to inject high quality current. LCL filter has better filtering effect, smaller system volume, and lower loss at the same cost compared with L filter [2]. However, as a result of inherent resonance of high-order filter, the circuit is prone to oscillating or even unstable outputs. [3] Therefore, extra requirements are put forward for the controller design for LCL-filtered inverter system. Additionally, 1-beat or 1.5-beat delay could be brought by discrete control and pulse width modulation [4]. The control delay will typically reduce the phase margin as well as complicate the design of the controller.Apart from the stability of the inverter system itself, another concern of the controller design is the interactive stability with the grid [5]. During actual operation, grid impedance is varying, resulting in stimulated resonances for the stable inverter system during independent operation and even undesirable tripping. Recently, the concept of passivity theory was applied to access the stability of grid-connected inverters in frequency domain, and many research works have been proposed to extend nonnegative real part of inverter output admittance [5–10]. It indicates that the resonance will not be stimulated if the interactive point falls in the passive region. Most control methods mainly consider passive regions up to the Nyquist frequency. However, passivity analysis and instability of inverter admittance above the Nyquist frequency should also be considered [8].Currently, most closed-loop control strategies rely on the classic linear control theory, which can be used to accurately design the controller parameters and evaluate the influence of the delay. These control methods are however difficult to fully eliminate interactive resonances under complex weak grid, and the traditional proportional integral control does not have sufficient disturbances rejection capability. Compared with the classical linear control theory, the modern nonlinear control theory has better robustness and performance, such as sliding mode control [11, 12], predictive control [13, 14], and passivity-based control (PBC) [15–26]. PBC has proven useful in the design of robust controllers for railway systems [19, 20], renewable generation systems [21], etc. In terms of the PBC design process, energy formation and damping injection are considered to be of obvious physical significance [17], and two branches are proposed by the Euler–Lagrange (EL) model and the port-controller Hamiltonian (PCH) model, called EL-PBC and IDA-PBC, respectively. [22–26] The modeling process of the EL-PBC method is simpler, but IDA-PBC offers more flexibility in the type of damping interconnection and the modeling process. Moreover, for the IDA-PBC, fast convergence speed of the error energy function can be determined by calculating the derivative of the closed-loop energy function and the injected damping [23]. For EL-PBC, when the error is large, the convergence speed is fast, and injected damping plays the main role. In contrast, the convergence speed becomes slower and the role of injected damping becomes weak when the error is small. Towards this end, IDA-PBC has been adopted in this study so as to achieve better performance under complex grid conditions.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 guideline and the try-and-error method is usually adopted. Therefore, for the purpose of assessing the stability of weak grid, the detailed design of digital controlled IDA-PBC is necessary. A brief summary of the main contributions is as follows.(1)
On the premise of ensuring Lyapunov stability, to simplify the design process for PCH model-based PBC, a more flexible interconnection matrix design method is used.(2)
This study demonstrates how to select the injected damping considering the effect of control delay using frequency-domain passivity theory based on linear control design.(3)
With the proposed controller parameters’ design method, the nonnegative real part of the LCL-filtered inverter output admittance can be achieved within switching frequency. If the passivity of subsystem is ensured, then the stability of the whole interlinked system in parallel form is guaranteed regardless of grid impedance.Moreover, in Section2, a brief description of the LCL-filtered inverter system is given, followed by the design of IDA-PBC methodology in Section 3. Instead of the original antisymmetric matrix, the interconnection matrix is simplified by using the upper triangular matrix with zero corner elements. In order to precisely design the injected damping gains, in Section 4, frequency-domain passivity theory is applied with the derived impedance model of IDA-PBC. In Section 5 and Section 6, system performance is demonstrated by simulations and experimental results. Furthermore, the study found that IDA-PBC also performed well in unbalanced grid. Section 7 concludes the study.


## 2. System Description and PCH Modeling

### 2.1. Description of the System

Figure1 depicts an inverter with a power supply voltage Udc and is filtered by a LCL filter including inverter-side inductor L1, filter capacitor C, and grid-side inductor L2. The parasitic resistances of L1 and L2 are represented by R1 and R2, respectively. The voltage vpcc is used for the synchronization of the grid current. Additionally, the inverter-side current i1, the capacitor voltage uC, and the grid-side current i2 should also be measured to control the injected current as well as suppress the resonances. It should be noted that the state observer or Kalman filter can be used to reduce the number of required sensors by estimating state variables [9, 11]. The equivalent grid reactance of Lg and Cg is connected to the grid at the point of common coupling (PCC), which can take a wide range of values. Therefore, the controller must be robust enough to withstand grid disturbance. Table 1 summarizes basic system parameters for analysis.Figure 1
Structure of LCL-filtered three-phase grid-connected inverter.Table 1
System parameters.
SymbolsComponentsValueUdcDC bus voltage350 VUgGrid voltage110V (RMS)foGrid frequency50 HzL1Inverter-side inductor1.2 mHCFilter capacitor6 uFL2Grid-side inductor1.2 mHfsSampling frequency10 kHzfswSwitching frequency10 kHz(1)L1di1kdt+R1i1k+uCk=uik,CduCkdt+i2k−i1k=0,L2di2kdt+R2i2k−uCk=−vpcck.In accordance with Figure1, the equations for a three-phase LCL filtered inverter system can be derived from Kirchhoff’s law in a stationary abc frame as follows:where k stands for the cases in abc coordinates. Through the rotation transformation, from (1), the mathematical model of the grid-connected inverter in d-q coordinates appears as(2)L1di1ddt=uid−R1i1d+ωL1i1q−uCd,L1di1qdt=uiq−R1i1q−ωL1i1d−uCq,CduCddt=i1d−i2d+ωCuCq,CduCqdt=i1q−i2q−ωCuCd,L2di2ddt=uCd−R2i2d+ωL2i2q−vpccd,L2di2qdt=uCq−R2i2q−ωL2i2d−vpccq.


### 2.2. PCH Modeling

In order to construct a robust controller for the above system, a PCH model with dissipation can be used [24]. It is possible to express the system’s PCH model as follows:(3)x˙=Jx−Rx∂Hx∂x+gxu,where J(x) represents the system interconnection properties and normally satisfy formula J(x) = -J(x)T, R(x) indicates the system dissipation characteristics, and H(x) is the Hamiltonian function; u indicates the exchange of energy between the outside world and the system. The coefficient matrix g(x) stands for the interconnections. The following is a definition of system variables:(4)x=x1x2x3x4x5x6T=L1i1dL1i1qCuCdCuCqL2i2dL2i2qT.The following is the expression for each of the vectors and matrices for the LCL-filter inverter according to (2):(5)J=0ωL1−1000−ωL100−100100ωC−1001−ωC00−100100ωL20001−ωL20,gx=1000000100000000−1000000−1T,(6)u=ud,uq,vpccd,vpccqT,R=diagR1R100R2R2.The stored energy in the LCL filter can be expressed as(7)Hx=x12+x222L1+x32+x422C+x52+x622L2.


## 2.1. Description of the System

Figure1 depicts an inverter with a power supply voltage Udc and is filtered by a LCL filter including inverter-side inductor L1, filter capacitor C, and grid-side inductor L2. The parasitic resistances of L1 and L2 are represented by R1 and R2, respectively. The voltage vpcc is used for the synchronization of the grid current. Additionally, the inverter-side current i1, the capacitor voltage uC, and the grid-side current i2 should also be measured to control the injected current as well as suppress the resonances. It should be noted that the state observer or Kalman filter can be used to reduce the number of required sensors by estimating state variables [9, 11]. The equivalent grid reactance of Lg and Cg is connected to the grid at the point of common coupling (PCC), which can take a wide range of values. Therefore, the controller must be robust enough to withstand grid disturbance. Table 1 summarizes basic system parameters for analysis.Figure 1
Structure of LCL-filtered three-phase grid-connected inverter.Table 1
System parameters.
SymbolsComponentsValueUdcDC bus voltage350 VUgGrid voltage110V (RMS)foGrid frequency50 HzL1Inverter-side inductor1.2 mHCFilter capacitor6 uFL2Grid-side inductor1.2 mHfsSampling frequency10 kHzfswSwitching frequency10 kHz(1)L1di1kdt+R1i1k+uCk=uik,CduCkdt+i2k−i1k=0,L2di2kdt+R2i2k−uCk=−vpcck.In accordance with Figure1, the equations for a three-phase LCL filtered inverter system can be derived from Kirchhoff’s law in a stationary abc frame as follows:where k stands for the cases in abc coordinates. Through the rotation transformation, from (1), the mathematical model of the grid-connected inverter in d-q coordinates appears as(2)L1di1ddt=uid−R1i1d+ωL1i1q−uCd,L1di1qdt=uiq−R1i1q−ωL1i1d−uCq,CduCddt=i1d−i2d+ωCuCq,CduCqdt=i1q−i2q−ωCuCd,L2di2ddt=uCd−R2i2d+ωL2i2q−vpccd,L2di2qdt=uCq−R2i2q−ωL2i2d−vpccq.


## 2.2. PCH Modeling

In order to construct a robust controller for the above system, a PCH model with dissipation can be used [24]. It is possible to express the system’s PCH model as follows:(3)x˙=Jx−Rx∂Hx∂x+gxu,where J(x) represents the system interconnection properties and normally satisfy formula J(x) = -J(x)T, R(x) indicates the system dissipation characteristics, and H(x) is the Hamiltonian function; u indicates the exchange of energy between the outside world and the system. The coefficient matrix g(x) stands for the interconnections. The following is a definition of system variables:(4)x=x1x2x3x4x5x6T=L1i1dL1i1qCuCdCuCqL2i2dL2i2qT.The following is the expression for each of the vectors and matrices for the LCL-filter inverter according to (2):(5)J=0ωL1−1000−ωL100−100100ωC−1001−ωC00−100100ωL20001−ωL20,gx=1000000100000000−1000000−1T,(6)u=ud,uq,vpccd,vpccqT,R=diagR1R100R2R2.The stored energy in the LCL filter can be expressed as(7)Hx=x12+x222L1+x32+x422C+x52+x622L2.


## 3. IDA-PBC Controller Design

The primary goal of the controller is to find the Hamiltonian energy storage function Ha(x), interconnection matrix Ja(x), and damping matrix Ra(x), which satisfy the following equation with a balance point at x∗:(8)Jdx−RdxKx=−Jax−Rax∂Hx∂x+gxu,where Kx=∂Hax/∂x, Jdx=Jx+Jax, Rdx=Rx+Rax, and Ha(x) is the undetermined function.Moreover, in order to ensure x∗ is locally stable equilibrium point, the following conditions should also be satisfied [15].(1)
In traditional design, strict structure preservation is provided(9)Jdx=Jx+Jax=−Jx+JaxT,Rdx=Rx+Rax=Rx+RaxT.(2)
K(x) is the integrated gradient of a scalar function:(10)∂Kx∂x=∂Kx∂xT.(3)
Expect the balance x∗; the desired dynamics is achieved if K(x) is satisfied:(11)Kx∗=−∂Hx∂x|x=x∗,(12)Hdx=Hx+Hax.(4)
At the balance point, K(x) should satisfy the following function based on the Lyapunov stability:(13)∂Kx∂x|x=x∗>−∂2Hx∂x2|x=x∗.For EL-PBC, J(x) is fixed and the damping is injected with constructed energy function. For IDA-PBC, it is more flexible to design Ja(x) and Ra(x) corresponding to desired energy function. At the stable equilibrium point x∗, the closed-loop system based on PCH is described as [22](14)x˙=Jdx−Rdx∂Hdx∂x.At the desired x∗, the tracking error of selected Hamiltonian function Hd(x) should be minimum. The derivation of the closed-loop energy function Hd(x) is depicted as(15)H˙dx=∂Hdx∂xTx˙=−∂Hdx∂xTRdx−Jdx∂Hdx∂x.Based on Lyapunov’s second criterion, the derivative of Hd(x) of the closed-loop energy function must be negative to attain an asymptotically stable equilibrium point. Conventional, Jd is designed as positive definite symmetric matrix based on structure preservation in (9). A more flexible method is proposed to design Jd in this study. The derivative of the energy function can be less than zero if [Rd-Jd] is a positive definite matrix. So, if injected damping Ra is positive, Jd can be designed as a lower triangular matrix or an upper triangular matrix with zero diagonal elements to simplify the design procedure, and it is also capable of ensuring that H˙dx<0. So, Ja, Jd, and Ra are defined as(16)Ja=0−ωL11000ωL100100000−ωC1000ωC00100000−ωL20000ωL20,Jd=000000000000100000010000001000000100,Ra=diagr1r1r2r2r3r3.The elements of Ra should be positive. Through the control states x5⟶x5∗ and x6⟶x6∗, the states x1, x2, x3, and x4 can gradually reach the equilibrium points, which are defined as follows:(17)x∗=L1i1d∗,L1i1d∗,CuCd∗,CuCq∗,L2i2d∗,L2i2q∗T.The derivative of Hd(x) can be defined as K(s) and corresponding components of K(s) can be viewed as follows:(18)Kx=k1x,k2x,k3x,k4x,k5x,k6xT,(19)k1x=−x1∗L1,k2x=−x2∗L1,k3x=−x3∗C,k4x=−x4∗C,k5x=−x5∗L2,k6x=−x6∗L2,When Ja, Ra, and K(x) are substituted to (1), reference values for the state variables of the system can be found in (20) which determined the passive controller u. As a result of (20), as shown in Figure 2, the cascaded three loops controller can be depicted. It is worth mention that “L1e,” “L2e,” “Ce,” “R1e,” and “R2e” are the initial filter parameters given in the controller, and the subscript “e” is used to differentiate with actual parameters of thr physical filter.Figure 2
Proposed IDA-PBC strategies for LCL-filtered grid-connected inverter.(20)ud∗=ucd−ωL1ei1q+r1i1d∗−i1d+R1ei1d∗,uq∗=ucq+ωL1ei1d+r1i1q∗−i1q+R1ei1q∗,i1d∗=i2d−ωCeuCq+r2uCd∗−uCd,i1q∗=i2q+ωCeuCd+r2uCq∗−uCq,uCd∗=vpccd−ωL2ei2q+r3i2d∗−i2d+R2ei2d,uCq∗=vpccq+ωL2ei2d+r3i2q∗−i2q+R2ei2q.By using (11) and (12), it is possible to calculate the amount of energy injected into the system by the controller:(21)Hax=−x1∗L1x1−x2∗L1x2−x3∗Cx3−x4∗Cx4−x5∗L2x5−x6∗L2x6.Accordingly, the closed-loop Hamiltonian energy function for this system is as follows:(22)Hdx=−x1∗L1x1−x2∗L1x2−x3∗Cx3−x4∗Cx4−x5∗L2x5−x6∗L2x6+x12+x222L1+x32+x422C+x52+x622L2.At the balanced point x∗, ∂Hdx/∂x is equal to 0, and the Hessian matrix of Hd(x) shown in (23) meets the requirement of (13):(23)∂2Hdx∂x2|x=x∗=diag1L11L11C1C1L21L2>0.Hence, Hd(x) has the minimum value at x∗. Furthermore,(24)H˙x=∂Hdx∂xTx˙=−∂Hdx∂xTRdx−Jdx∂Hdx∂x≤0.And apart from x∗, there are no solutions of x (t) which can stay in the following solutions set:(25)x|x∈Rn,∂THd∂xRd−Jd∂Hd∂x=0.From the LaSalle invariant set theorem [22], the equilibrium point x∗ is considered asymptotically stable. The energy exchange with the network has been included in the energy function of the closed-loop system. When the system is dissipative, this also implies that it is passive. In order to select the values of the injected damping coefficients, Section 4 will analyze the passivity of the closed-loop output admittance using the impedance model.


## 4. Impedance Model and Stability Oriented Design

### 4.1. Impedance Modeling

With the traditional design of the IDA-PBC controller outlined, the digital control is ignored. The system is modeled with equivalent Laplace transform for the controller equations to select the appropriate parameters of injected damping gains considering the effect of discretization delay. Based on (20), it is possible to describe the IDA-PBC controller as a three-stage controller for LCL-filtered inverters in the following matrices:(26)ud∗uq∗=E00Ei1d∗i1q∗−r1ωL1e−ωL1er1i1di1q+uCduCq,(27)i1d∗i1q∗=F00FuCd∗uCq∗−r2ωCe−ωCer2uCduCq+i2di2q,(28)uCd∗uCq∗=G00Gi2d∗i2q∗−r3ωL2e−ωL2er3i2di2q+vpccdvpccq,where E=R1e+r1, F=r2, and G=R2e+r3.In Figure2, the delay in the system is added after the deduced reference control law, resulting in the following expression [8]:(29)Gds=sinωT/2ωT/2e−1.5Ts,where T represents the sampling period. The delay Gd(s) is written as Gd to simplify the expression in the following. Then, the control signal is sent to the LCL filter-based inverter depicted in Figure 2:(30)uduq=Gdud∗uq∗.The LCL-filtered inverter system model is expressed as follows:(31)i2di2q=0ωL2−ωL20A00Ai2di2q+A00AuCduCq−A00Avpccdvpccq,(32)uCduCq=0ωC−ωC0B00BuCduCq+B00Bi1di1q−B00Bi2di2q,(33)i1di1q=0ωL1−ωL10D00Di1di1q+D00Duduq−D00DuCduCq,where A=1/sL2+R2, B=1/sC, and D=1/sL1+R1.Substituting (30) to (33), ud and uq can be canceling, and it can be derived that(34)1−ωL1DωL1D1i1di1q=D×Gd00D×Gdud∗uq∗−D00DuCduCq.Then, substituting (33) to (26), ud∗ and uq∗ can be canceling, and it can be obtained that(35)1−ωL1DωL1D1i1di1q=DE×Gd00DE×Gdi1d∗i1q∗−r1D×GdωL1eD×Gd−ωL1eD×Gdr1D×Gdi1di1q+DGd−100DGd−1uCduCq.Based on (34), a description of the inner loop response would be as follows:(36)i1di1q=Xiψi−ψiXii1d∗i1q∗−Yi−φiφiYiuCduCq,where ψi and φi are the coupling components d-q axis and Xi and Yi correspond to the transfer function of closed-loop and output admittance of the inner loop, respectively:(37)Xi=DE×Gd1+r1D×Gd1+r1D×Gd2+ωL1D−ωL1eD×Gd2,(38)Yi=−DGd−11+r1D×Gd1+r1D×Gd2+ωL1D−ωL1eD×Gd2.Similarly, combining functions (26)-(33), the equivalent response of the overall controller can be expressed in (39). The derivation process is omitted here:(39)i2d∗i2q∗=Goψ−ψGoi2di2q−Yo−φφYovpccdvpccq,where ψ and φ are the coupling components on the d-q axis and Go and Yo represent the closed-loop transfer function and the inverter output admittance of the system, respectively.


### 4.2. Proposed Parameters’ Design Procedure

As can be seen from Figure2, the IDA-PBC controller offers a three-loop structure for the LCL inverter system that incorporates three gains (r1, r2, r3) that must be determined. Traditional IDA-PBCs can achieve stable equilibrium conditions with energy shaping techniques. However, the design of interconnected damping gains is not guided by any specific design procedure. The following is a step-by-step design procedure for IDA-PBC gains that aims for fast dynamic response and high level of robustness against external disturbances.


#### 4.2.1. Design of the Inner Loop Gain r1

As the gain of the inner loop, r1 should be designed to achieve a fast response. For convenience, a first-order inertial model is substituted for the delay in the design of inner loop. Thus, the closed-loop transfer function of the inner loop is expressed in function (40):(40)i1i1∗=R1+r11.5Ts+1sL1+R1+r1,(41)r1≈L16ξ2T,ξ=1.It can be seen from (40) that the inner loop is a second-order system. According to classical control theory, the critically damped case is preferred to reach shortest time for the system. Since the critical damping ratio is equal to 1, then r1 is chosen as 2. It is should, however, be noted that PI regulator can be applied instead of the damping r1 with proper integral coefficient to remove the steady-state error.


#### 4.2.2. Design of the Middle Loop Gain r2 and the Outer Loop Gain r3

Here, design of r2 and r3 relies on passivity of the inverter system impedance with control delay to ensure a stable system. Based on passivity theory, within switching frequency, if positive real part of system output impedance can be ensured, the system is passive [5]. For the analysis, mathematics software can be applied to calculate closed-loop transfer function Go(s) and output admittance Yo(s) for the outer loop. In this process, the delay function is substituted by Euler’s formula. During the steady state condition, i2d/i2d∗ = 1 and i2d/i2q = 0. Based on function (39), ignoring the coupling items, Go(s) is expressed as follows:After deciding on r1 = 2, the relationship between frequency, the real part of the inverter output admittance, and r2 or r3 can be plotted. The value of r2 is not independent, but varies with value of r3. Assume that r3 is ensured and then determine the approximate range of r2. Likewise, select the middle value of r2 within the stable range, and determine the stable range of r1 again. In addition, the system closed-loop poles are drawn with different r1 and r2. This is demonstrated in Figures 3 and 4, where two groups of r1 and r2 are selected for verification.Figure 3
Real part of LCL-filtered inverter output admittance with IDA-PBC when (a) r3 = 25 and r2 varies, (b) r2 = 0.15 and r3 varies, (c) r3 = 10 and r2 varies, and (d) r2 = 0.1 and r3 varies.
(a)(b)(c)(d)(42)i2di2d∗=XBFG1−BY+Br2XA−1+XBFr3+B−XB.Figure 4
Closed-loop zero-pole maps of LCL-filtered inverter with IDA-PBC when (a) r3 = 25 and r2 varies, (b) r2 = 0.15 and r3 varies, (c) r3 = 10 and r2 varies, and (d) r2 = 0.1 and r3 varies.
(a)(b)(c)(d)The relationship between r2, inverter output admittance, and frequency is illustrated in Figure 3(a), assuming that r3 is 25. In order to ensure positive real part of Yo(s), r2 should be less than 0.19, which can also be proved with the closed-loop zero-pole maps shown in Figure 4(a).Then, assuming that r2 is 0.15, reverse-seek the range of r3, as demonstrated in Figures 3(b) and 4(b). In the second round, r3 is selected as 10, and the stable range of r2 are illustrated in Figures 3(c) and 4(c). Then, the range of r3 can be reached by reverse-seeking, assuming that r2 is 0.1 and r1 is 2, as shown in Figure 3(d). After the iterations, r2 and r3 are selected as 0.15 and 15, consequently. The overall flow diagram of the proposed design procedure for LCL-filtered grid-connected inverter with IDA-PBC is shown in Figure 5.Figure 5
Flow diagram of the proposed design procedure for LCL-filtered grid-connected inverter with IDA-PBC.According to the figures of closed-loop pole-zero maps and output admittance real parts, the calculated ranges of parameters based on the internal stability of the system basically coincide with the ranges calculated according to the passivity of the output admittance. Due to the fact that the energy function of the closed-loop system represents the physical state of energy at the equilibrium point and accounts for external energy input in its entirety, with the selected parameters, the real part of the output admittance can be always positive and passive within switching frequency. Hence, the overall system can remain stable regardless of the variation of the grid impedance.


## 4.1. Impedance Modeling

With the traditional design of the IDA-PBC controller outlined, the digital control is ignored. The system is modeled with equivalent Laplace transform for the controller equations to select the appropriate parameters of injected damping gains considering the effect of discretization delay. Based on (20), it is possible to describe the IDA-PBC controller as a three-stage controller for LCL-filtered inverters in the following matrices:(26)ud∗uq∗=E00Ei1d∗i1q∗−r1ωL1e−ωL1er1i1di1q+uCduCq,(27)i1d∗i1q∗=F00FuCd∗uCq∗−r2ωCe−ωCer2uCduCq+i2di2q,(28)uCd∗uCq∗=G00Gi2d∗i2q∗−r3ωL2e−ωL2er3i2di2q+vpccdvpccq,where E=R1e+r1, F=r2, and G=R2e+r3.In Figure2, the delay in the system is added after the deduced reference control law, resulting in the following expression [8]:(29)Gds=sinωT/2ωT/2e−1.5Ts,where T represents the sampling period. The delay Gd(s) is written as Gd to simplify the expression in the following. Then, the control signal is sent to the LCL filter-based inverter depicted in Figure 2:(30)uduq=Gdud∗uq∗.The LCL-filtered inverter system model is expressed as follows:(31)i2di2q=0ωL2−ωL20A00Ai2di2q+A00AuCduCq−A00Avpccdvpccq,(32)uCduCq=0ωC−ωC0B00BuCduCq+B00Bi1di1q−B00Bi2di2q,(33)i1di1q=0ωL1−ωL10D00Di1di1q+D00Duduq−D00DuCduCq,where A=1/sL2+R2, B=1/sC, and D=1/sL1+R1.Substituting (30) to (33), ud and uq can be canceling, and it can be derived that(34)1−ωL1DωL1D1i1di1q=D×Gd00D×Gdud∗uq∗−D00DuCduCq.Then, substituting (33) to (26), ud∗ and uq∗ can be canceling, and it can be obtained that(35)1−ωL1DωL1D1i1di1q=DE×Gd00DE×Gdi1d∗i1q∗−r1D×GdωL1eD×Gd−ωL1eD×Gdr1D×Gdi1di1q+DGd−100DGd−1uCduCq.Based on (34), a description of the inner loop response would be as follows:(36)i1di1q=Xiψi−ψiXii1d∗i1q∗−Yi−φiφiYiuCduCq,where ψi and φi are the coupling components d-q axis and Xi and Yi correspond to the transfer function of closed-loop and output admittance of the inner loop, respectively:(37)Xi=DE×Gd1+r1D×Gd1+r1D×Gd2+ωL1D−ωL1eD×Gd2,(38)Yi=−DGd−11+r1D×Gd1+r1D×Gd2+ωL1D−ωL1eD×Gd2.Similarly, combining functions (26)-(33), the equivalent response of the overall controller can be expressed in (39). The derivation process is omitted here:(39)i2d∗i2q∗=Goψ−ψGoi2di2q−Yo−φφYovpccdvpccq,where ψ and φ are the coupling components on the d-q axis and Go and Yo represent the closed-loop transfer function and the inverter output admittance of the system, respectively.


## 4.2. Proposed Parameters’ Design Procedure

As can be seen from Figure2, the IDA-PBC controller offers a three-loop structure for the LCL inverter system that incorporates three gains (r1, r2, r3) that must be determined. Traditional IDA-PBCs can achieve stable equilibrium conditions with energy shaping techniques. However, the design of interconnected damping gains is not guided by any specific design procedure. The following is a step-by-step design procedure for IDA-PBC gains that aims for fast dynamic response and high level of robustness against external disturbances.


### 4.2.1. Design of the Inner Loop Gain r1

As the gain of the inner loop, r1 should be designed to achieve a fast response. For convenience, a first-order inertial model is substituted for the delay in the design of inner loop. Thus, the closed-loop transfer function of the inner loop is expressed in function (40):(40)i1i1∗=R1+r11.5Ts+1sL1+R1+r1,(41)r1≈L16ξ2T,ξ=1.It can be seen from (40) that the inner loop is a second-order system. According to classical control theory, the critically damped case is preferred to reach shortest time for the system. Since the critical damping ratio is equal to 1, then r1 is chosen as 2. It is should, however, be noted that PI regulator can be applied instead of the damping r1 with proper integral coefficient to remove the steady-state error.


### 4.2.2. Design of the Middle Loop Gain r2 and the Outer Loop Gain r3

Here, design of r2 and r3 relies on passivity of the inverter system impedance with control delay to ensure a stable system. Based on passivity theory, within switching frequency, if positive real part of system output impedance can be ensured, the system is passive [5]. For the analysis, mathematics software can be applied to calculate closed-loop transfer function Go(s) and output admittance Yo(s) for the outer loop. In this process, the delay function is substituted by Euler’s formula. During the steady state condition, i2d/i2d∗ = 1 and i2d/i2q = 0. Based on function (39), ignoring the coupling items, Go(s) is expressed as follows:After deciding on r1 = 2, the relationship between frequency, the real part of the inverter output admittance, and r2 or r3 can be plotted. The value of r2 is not independent, but varies with value of r3. Assume that r3 is ensured and then determine the approximate range of r2. Likewise, select the middle value of r2 within the stable range, and determine the stable range of r1 again. In addition, the system closed-loop poles are drawn with different r1 and r2. This is demonstrated in Figures 3 and 4, where two groups of r1 and r2 are selected for verification.Figure 3
Real part of LCL-filtered inverter output admittance with IDA-PBC when (a) r3 = 25 and r2 varies, (b) r2 = 0.15 and r3 varies, (c) r3 = 10 and r2 varies, and (d) r2 = 0.1 and r3 varies.
(a)(b)(c)(d)(42)i2di2d∗=XBFG1−BY+Br2XA−1+XBFr3+B−XB.Figure 4
Closed-loop zero-pole maps of LCL-filtered inverter with IDA-PBC when (a) r3 = 25 and r2 varies, (b) r2 = 0.15 and r3 varies, (c) r3 = 10 and r2 varies, and (d) r2 = 0.1 and r3 varies.
(a)(b)(c)(d)The relationship between r2, inverter output admittance, and frequency is illustrated in Figure 3(a), assuming that r3 is 25. In order to ensure positive real part of Yo(s), r2 should be less than 0.19, which can also be proved with the closed-loop zero-pole maps shown in Figure 4(a).Then, assuming that r2 is 0.15, reverse-seek the range of r3, as demonstrated in Figures 3(b) and 4(b). In the second round, r3 is selected as 10, and the stable range of r2 are illustrated in Figures 3(c) and 4(c). Then, the range of r3 can be reached by reverse-seeking, assuming that r2 is 0.1 and r1 is 2, as shown in Figure 3(d). After the iterations, r2 and r3 are selected as 0.15 and 15, consequently. The overall flow diagram of the proposed design procedure for LCL-filtered grid-connected inverter with IDA-PBC is shown in Figure 5.Figure 5
Flow diagram of the proposed design procedure for LCL-filtered grid-connected inverter with IDA-PBC.According to the figures of closed-loop pole-zero maps and output admittance real parts, the calculated ranges of parameters based on the internal stability of the system basically coincide with the ranges calculated according to the passivity of the output admittance. Due to the fact that the energy function of the closed-loop system represents the physical state of energy at the equilibrium point and accounts for external energy input in its entirety, with the selected parameters, the real part of the output admittance can be always positive and passive within switching frequency. Hence, the overall system can remain stable regardless of the variation of the grid impedance.


## 4.2.1. Design of the Inner Loop Gain r1

As the gain of the inner loop, r1 should be designed to achieve a fast response. For convenience, a first-order inertial model is substituted for the delay in the design of inner loop. Thus, the closed-loop transfer function of the inner loop is expressed in function (40):(40)i1i1∗=R1+r11.5Ts+1sL1+R1+r1,(41)r1≈L16ξ2T,ξ=1.It can be seen from (40) that the inner loop is a second-order system. According to classical control theory, the critically damped case is preferred to reach shortest time for the system. Since the critical damping ratio is equal to 1, then r1 is chosen as 2. It is should, however, be noted that PI regulator can be applied instead of the damping r1 with proper integral coefficient to remove the steady-state error.


## 4.2.2. Design of the Middle Loop Gain r2 and the Outer Loop Gain r3

Here, design of r2 and r3 relies on passivity of the inverter system impedance with control delay to ensure a stable system. Based on passivity theory, within switching frequency, if positive real part of system output impedance can be ensured, the system is passive [5]. For the analysis, mathematics software can be applied to calculate closed-loop transfer function Go(s) and output admittance Yo(s) for the outer loop. In this process, the delay function is substituted by Euler’s formula. During the steady state condition, i2d/i2d∗ = 1 and i2d/i2q = 0. Based on function (39), ignoring the coupling items, Go(s) is expressed as follows:After deciding on r1 = 2, the relationship between frequency, the real part of the inverter output admittance, and r2 or r3 can be plotted. The value of r2 is not independent, but varies with value of r3. Assume that r3 is ensured and then determine the approximate range of r2. Likewise, select the middle value of r2 within the stable range, and determine the stable range of r1 again. In addition, the system closed-loop poles are drawn with different r1 and r2. This is demonstrated in Figures 3 and 4, where two groups of r1 and r2 are selected for verification.Figure 3
Real part of LCL-filtered inverter output admittance with IDA-PBC when (a) r3 = 25 and r2 varies, (b) r2 = 0.15 and r3 varies, (c) r3 = 10 and r2 varies, and (d) r2 = 0.1 and r3 varies.
(a)(b)(c)(d)(42)i2di2d∗=XBFG1−BY+Br2XA−1+XBFr3+B−XB.Figure 4
Closed-loop zero-pole maps of LCL-filtered inverter with IDA-PBC when (a) r3 = 25 and r2 varies, (b) r2 = 0.15 and r3 varies, (c) r3 = 10 and r2 varies, and (d) r2 = 0.1 and r3 varies.
(a)(b)(c)(d)The relationship between r2, inverter output admittance, and frequency is illustrated in Figure 3(a), assuming that r3 is 25. In order to ensure positive real part of Yo(s), r2 should be less than 0.19, which can also be proved with the closed-loop zero-pole maps shown in Figure 4(a).Then, assuming that r2 is 0.15, reverse-seek the range of r3, as demonstrated in Figures 3(b) and 4(b). In the second round, r3 is selected as 10, and the stable range of r2 are illustrated in Figures 3(c) and 4(c). Then, the range of r3 can be reached by reverse-seeking, assuming that r2 is 0.1 and r1 is 2, as shown in Figure 3(d). After the iterations, r2 and r3 are selected as 0.15 and 15, consequently. The overall flow diagram of the proposed design procedure for LCL-filtered grid-connected inverter with IDA-PBC is shown in Figure 5.Figure 5
Flow diagram of the proposed design procedure for LCL-filtered grid-connected inverter with IDA-PBC.According to the figures of closed-loop pole-zero maps and output admittance real parts, the calculated ranges of parameters based on the internal stability of the system basically coincide with the ranges calculated according to the passivity of the output admittance. Due to the fact that the energy function of the closed-loop system represents the physical state of energy at the equilibrium point and accounts for external energy input in its entirety, with the selected parameters, the real part of the output admittance can be always positive and passive within switching frequency. Hence, the overall system can remain stable regardless of the variation of the grid impedance.


## 5. Simulation Results

Simulations were conducted with selected system parameter values from Table1 and controller parameters from Table 2 in order to evaluate the effectiveness of the proposed strategy.Table 2
Controller parameters.
SymbolsComponentsValuer1Inner loop damping2r2Middle loop damping0.15r3Loop damping15


### 5.1. Under Balanced Grid Voltage

Figure6 depicts the waveforms results of system when implementing the proposed IDA-PBC strategy under the steady state. The grid impedance is zero for 0 < t<0.06s, and Lg is set as 3.6mH for 0.06s < t<0.12s, and then, Lg = 3.6mH and Cg = 4uF for 0.12s < t<0.18s. The inductor and the capacitor are added in the grid at 0.6s and 0.12s. The system returns to the normal state after one period. The 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-PBC strategy. It can be seen that traditional IDA-PBC strategy can be stable in strong grid and inductive grid, but it is unstable under some capacitive grid impedance cases. The proposed IDA-PBC strategy is more robust under complex grid impedance condition.Figure 6
Simulated results when grid impedance varies with proposed IDA-PBC strategy.Figure 7
Simulated results when grid impedance varies with traditional IDA-PBC strategy.The performance of the proposed method during a transient response is also tested under strong grid, inductive grid, and complex grid, as illustrated in Figure8. 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.Figure 8
Simulated results of transient responses of grid currents under balanced grid when (a) Lg = 0 mH and Cg = 0 uF, (b) Lg = 4.8 mH and Cg = 0 uF, and (c) Lg = 4.8 mH and Cg = 3 uF.
(a)(b)(c)


### 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. Figure9 shows the results of injected currents when vga and vgb drop by 36% and 18%, with vga = 70(RMS) and vgb = 90(RMS), respectively. It is apparent that grid currents are still sinusoidal and balanced with the tracked reference value under unbalanced grid.Figure 9
Simulated results of grid voltages and grid currents when grid impedance varies under unbalanced grid.


### 5.3. Grid Impedance Variation in a Wide Range

Figure10 illustrates a simulation of grid currents to demonstrate the robustness of the proposed control algorithm under a wide range of grid impedance variations when Cg is fixed as 7uF and Lg varies from 0 to 10mH within 0.1s. Figure 11 depicts the corresponding results of the grid currents when Lg is set as 3mH and Cg varies from 0 to 15uF within 0.1s. It is clear that the system can inject high quality and stable currents both in inductive and capacitive grid in a wide range with the selected parameters for the IDA-PBC method.Figure 10
Simulated results of grid currents when Cg = 7 uF; Lg varies from 0 mH to 10mH.Figure 11
Simulated results of grid currents when Lg = 3 mH; Cg varies from 0 uF to 15 uF.


## 5.1. Under Balanced Grid Voltage

Figure6 depicts the waveforms results of system when implementing the proposed IDA-PBC strategy under the steady state. The grid impedance is zero for 0 < t<0.06s, and Lg is set as 3.6mH for 0.06s < t<0.12s, and then, Lg = 3.6mH and Cg = 4uF for 0.12s < t<0.18s. The inductor and the capacitor are added in the grid at 0.6s and 0.12s. The system returns to the normal state after one period. The 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-PBC strategy. It can be seen that traditional IDA-PBC strategy can be stable in strong grid and inductive grid, but it is unstable under some capacitive grid impedance cases. The proposed IDA-PBC strategy is more robust under complex grid impedance condition.Figure 6
Simulated results when grid impedance varies with proposed IDA-PBC strategy.Figure 7
Simulated results when grid impedance varies with traditional IDA-PBC strategy.The performance of the proposed method during a transient response is also tested under strong grid, inductive grid, and complex grid, as illustrated in Figure8. 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.Figure 8
Simulated results of transient responses of grid currents under balanced grid when (a) Lg = 0 mH and Cg = 0 uF, (b) Lg = 4.8 mH and Cg = 0 uF, and (c) Lg = 4.8 mH and Cg = 3 uF.
(a)(b)(c)


## 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. Figure9 shows the results of injected currents when vga and vgb drop by 36% and 18%, with vga = 70(RMS) and vgb = 90(RMS), respectively. It is apparent that grid currents are still sinusoidal and balanced with the tracked reference value under unbalanced grid.Figure 9
Simulated results of grid voltages and grid currents when grid impedance varies under unbalanced grid.


## 5.3. Grid Impedance Variation in a Wide Range

Figure10 illustrates a simulation of grid currents to demonstrate the robustness of the proposed control algorithm under a wide range of grid impedance variations when Cg is fixed as 7uF and Lg varies from 0 to 10mH within 0.1s. Figure 11 depicts the corresponding results of the grid currents when Lg is set as 3mH and Cg varies from 0 to 15uF within 0.1s. It is clear that the system can inject high quality and stable currents both in inductive and capacitive grid in a wide range with the selected parameters for the IDA-PBC method.Figure 10
Simulated results of grid currents when Cg = 7 uF; Lg varies from 0 mH to 10mH.Figure 11
Simulated results of grid currents when Lg = 3 mH; Cg varies from 0 uF to 15 uF.


## 6. Experimental Validation

To further validate the effectiveness of the IDA-PBC for LCL-filtered inverter with selected parameters, an experimental setup of 3-KW/110-V/three-phase grid inverter was built to test. The dc-link source to the Danfoss FC302 inverter is provided by Chroma dc power supply. The 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 Table1, the selected parameters of r1, r2, and r3 of the IDA-PBC controller are 2, 0.15, and 15, respectively. Based on the simulation results, it is clear that the system is capable of performing optimally in a balanced grid. The experimental results of the proposed controller are displayed here under the unbalanced voltage condition. The voltage of phase A is emulated by a 36% dips with vga = 70V (RMS).A study of experimental grid currents and voltages under the steady state is presented in Figures12 and 13. The grid currents can be controlled to be nearly sinusoidal with the proposed method in both inductive and capacitive 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.Figure 12
Experimental results of grid voltages and currents with Lg = 3.6 mH and Cg = 0 uF.Figure 13
Experimental results of grid voltages and currents with Lg = 3.6 mH and Cg = 6 uF.Figure 14
Experimental results of transient responses of grid currents with Lg = 0 mH and Cg = 0 uF.Figure 15
Experimental results of transient responses of grid currents with Lg = 3.6 mH and Cg = 0 uF.Figure 16
Experimental results of transient responses of grid currents with Lg = 3.6 mH and Cg = 6 uF.The grid current can rapidly track the reference without 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 response time is less than 3 ms.


## 7. Conclusion

A modified IDA-PBC strategy and design method for calculating damping gains based on an impedance model are presented in this study. Simulations and experimental tests have been used to determine the effectiveness of the proposed controller, from which the following conclusion can be drawn.(1)
The stability of IDA-PBC with the assignment of the desired energy function to the closed-loop system is coinciding to the passivity of the closed-loop system output admittance. It is a good feature to allow this control method to be widely used in weak grid. The optional range of injected damping parameters can be selected based on the principle to satisfy the output impedance is passive.(2)
The design procedure of IDA-PBC considers the effect of control delay and a more flexible interconnection matrix design method is proposed. With the proposed IDA-PBC strategy, the inverter system is passive within the switching frequency, which allows high quality current to be injected to the grid, regardless of large variations in grid impedance and unbalanced grid voltages.



---

*Source: 7515321-2022-04-15.xml*