Linear Quadratic Regulator Optimal Control with Integral Action (LQRIC) for LC-Coupling Hybrid Active Power Filter

Abstract

Renewable energy generation and nonlinear load devices will generate harmonics and reactive power to power grids, resulting in current distortion and low power factors. To solve the power quality problems, the LC-coupling hybrid active power filter (LC-HAPF) is proposed, with lower DC-link voltage and lower cost compared with conventional active power filters (APFs). The LC-HAPF requires a controller to operate, therefore, hysteresis current control (HCC) and proportional current control (PCC) were proposed. However, they both result in significant steady-state error. Hence, linear quadratic regulator control (LQRC) with integral action (LQRIC) is proposed for the LC-HAPF in this paper to mitigate the steady-state error. The d-q-0 coordinate state-space model of the LQRIC-controlled LC-HAPF is derived, and a detailed design guideline of the weighting matrices Q and R of LQRIC is given. By the state-space model and weighting matrices, the gain matrix K of LQRIC can be acquired by MATLAB, thus a good steady-state performance can be ensured. Finally, the simulation results of different controllers for the LC-HAPF under 40V and 50V DC-link voltages are given to verify the effectiveness of the proposed LQRIC. The experimental results of LQRIC-controlled LC-HAPF are also given to verify the feasibility of the proposed LQRIC.

1. Introduction

In recent years, clean and renewable energy, such as solar power and wind power, has been vigorously promoted and applied [1,2,3]. However, the photovoltaic (PV) systems [1] and typical wind power systems [2] require inverters to operate. They cause power quality issues such as a low power factor and current harmonic pollution, which result in the failure of electrical equipment, the interference of communication circuits, etc., bringing economic losses to societies [4,5].

In order to solve the power quality problem, capacitor bank (CB) was proposed and used for reactive power compensation in the 1900s. In the 1940s, passive power filter (PPF) was proposed to achieve both reactive power and harmonic compensation [6]. However, it can only compensate fixed harmonic frequency and suffers from resonance problems. Therefore, the active power filter (APF), which combines the voltage source inverter (VSI) and inductor, was proposed in 1976 [7]. It overcomes the shortcomings of CB and PPF, and can achieve dynamic compensation. However, APF has the problem of high cost due to the high DC-link voltage. The hybrid active power filter (HAPF) was proposed to overcome the disadvantages of PPF and APF. It can reduce the operation cost by reducing the DC-link voltage [8,9,10]. There are different topologies of HAPF. Among them, the LC-coupling hybrid active power filter (LC-HAPF), which has an LC branch and a voltage source inverter (VSI) in series, was proposed in 2003. It can achieve harmonic current and reactive power compensation [8,9,10,11]. The thyristor-controlled LC-coupling hybrid active power filter (TCLC-HAPF) was proposed in 2014 [12], and can provide a wide range of reactive power compensation for both capacitive and inductive loads in the medium-voltage-level power system. However, it also increases the control complexity and power consumption. Next, LCLC-coupling hybrid active power filter (LCLC-HAPF) was proposed in 2020 [13]. It has a good high-order harmonic attenuation characteristic and lower inductor cost. However, it has stability problems and needs the assistance of active damping control. Therefore, considering the complexity and the system stability, the LC-HAPF is mainly focused on in this paper.

In addition to the topology of the LC-HAPF, the controller design also plays an important role in the compensation performance. Hysteresis current control (HCC) is the most widely used control method and provides fast transient response and acceptable steady-state performance [14]. However, due to its varied switching frequency, it generates harmonic current in a wide frequency range, resulting in a large output current ripple. Thus, the classical fixed switching frequency control, proportional current control (PCC), is proposed [15] to mitigate the current ripple for a better current tracking ability comparing with the HCC. However, the control gain optimization of PCC requires a trial-and-error process, therefore retaining a certain degree of steady-state error.

To obtain a lower steady-state error, a linear quadratic regulator control (LQRC) [16,17,18,19,20,21,22,23] was proposed for APF, HAPF and other applications due to the good steady-state performance. Moreover, some research suggests that LQRC with integral action (LQRIC) [24,25,26,27] is able to achieve a better steady-state performance because of the integral term. Therefore, this paper proposes the LQRIC for the LC-HAPF to achieve superior steady-state performance, and the LQRC for the LC-HAPF for the comparative study. Moreover, the gain matrix K of LQRC and LQRIC is obtained from the cost function of state errors, weighting matrices and system state space model. It results in a better steady-state performance than the PCC, whose gain optimization is just the trial-and-error process.

The objective of this paper is to design a LQRIC for the LC-HAPF to improve the steady-state performance under low DC-link voltage. Figure 1 is the graphical abstract of this paper.

The contributions of this paper are summarized as follows:

(1)

A LQRIC for the three-phase four-wire LC-HAPF is proposed to obtain a superior compensation performance.

(2)

Design the state-space model in d-q-0 coordinate of a LQRIC-controlled LC-HAPF.

(3)

Study the design of the weighting matrices, Q and R, of the LQRIC for LC-HAPF to ensure a good compensation performance.

(4)

Compare the simulation results with HCC, PCC and LQRC under different DC-link voltage conditions and verify the effectiveness of the proposed LQRIC for the LC-HAPF.

(5)

The experimental results for LQRIC under different DC-link voltages are given to verify the feasibility of the proposed LQRIC.

The arrangement of this paper is shown below. Section 2 establishes the system configuration of the three-phase four-wire LC-HAPF, and also gives the system model and introduces the parameter design. Then, Section 3 introduces the design of the proposed LQRC and LQRIC. The state-space model of LC-HAPF in d-q-0 coordinate and the design method of the weighting matrices, Q and R, are also given. Section 4 provides the overall control block diagram of the different controllers for the LC-HAPF. Section 5 provides the simulation results for verifying the effectiveness of the proposed LQRIC. Section 6 provides the experimental results for verifying the feasibility of the proposed LQRIC. The overall performance of different controllers is discussed in Section 7. Finally, the paper is concluded in Section 8.

2. Circuit Configuration of LC-HAPF

2.1. Circuit Configuration and System Modeling

Figure 2 shows a three-phase four-wire LC-HAPF with a balanced nonlinear load [9,10,11]. The subscript x indicates the phase a, b, c and n. v_sx, i_sx, v_x and i_Lx are the source voltage, source current, the point of common coupling (PCC) voltage, and load current for each phase, respectively. i_cx, v_cx and v_invx are the compensation current, compensation capacitor voltage and inverter output voltage, respectively. L_Lx, R_Lx and C_Lx are the inductor, resistor and capacitor of the nonlinear loads, which are the full-bridge rectifier. L_cx, C_cx and R_cx are the inductor, capacitor and equivalent resistance of the coupling PPF. T_1x and T_2x are the trigger signals for IGBTs. The upper and lower DC-link capacitor voltage, V_dcU and V_dcL, satisfy V_dcU = V_dcL = 0.5V_dc, where V_dc is the DC-link voltage of VSI.

The PPF and VSI consist of LC-HAPF, which generate i_cx to compensate the grid harmonics and reactive power. The differential equations for LC-HAPF are set up by (1).

$$\{\begin{array}{l}{L}_{cx}\frac{d{i}_{cx}}{dt}=v_{invx}-v_x+v_{cx}-i_{cx}{R}_{cx}\\ C_{cx}\frac{d{v}_{cx}}{dt}=i_{cx}\end{array}$$

(1)

Based on (1), the state-space model of LC-HAPF is set up by (2).

$$\{\begin{array}{l}\frac{d}{dt}\mathrm{e}=\mathrm{Ae}+\mathrm{Bu}+\mathrm{Ez}\\ \mathrm{y}=\mathrm{Ce}+\mathrm{Du}\end{array}$$

(2)

where e, u, z and y indicate state, control, disturbance and output variable matrix, and they are expressed as $\mathrm{e}={\left[\begin{array}{ccc}{i}_{ca}& i_{cb}& i_{cc}\end{array}\right]}^T$, $\mathrm{u}={\left[\begin{array}{ccc}{v}_{inva}& v_{invb}& v_{invc}\end{array}\right]}^T$, $\mathrm{z}={\left[\begin{array}{ccc}{v}_a-v_{ca}& v_b-v_{cb}& v_c-v_{cc}\end{array}\right]}^T$ and $\mathrm{y}={\left[\begin{array}{ccc}{i}_{ca}& i_{cb}& i_{cc}\end{array}\right]}^T$. T indicates the transpose of the matrix.

The upper equation in (2) is the system state differential equation of the LC-HAPF, the matrix A, B and E are state, control and disturbance matrix. The lower equation in (2) is the output equation of the LC-HAPF, which includes the output state matrix, C, and the output control matrix, D. The value of the A, B, C, D and E matrices for LQRC and LQRIC will be discussed in Section 3.

2.2. Parameter Design

The value design of L_cx and C_cx are based on the root-mean-square of PCC voltage V_{x_rms} and the reactive power for the load Q_Lx in (3), where X_Cc and X_Lc are the reactance of L_cx and C_cx and (4) shows the relation between L_cx and C_cx. ω₁ is the fundamental angular frequency, ω_m is the harmonic angular frequency and the subscript m is the harmonic order.

$$\frac{V_{x\_rms}^2}{\left|Q_{Lx}\right|}=X_{Cc}-X_{Lc}=\frac{1}{{\omega }_1{C}_{cx}}-{\omega }_1{L}_{cx}$$

(3)

$$L_{cx}=\frac{1}{{\omega }_m{}^2{C}_{cx}}$$

(4)

The full-bridge rectifier load will generate odd order (m = 3, 5, 7, …) harmonic. As the main harmonic current of full bridge rectifier are the 3rd and 5th orders [28,29], in this paper, the 5th order harmonic is chosen to reduce the structure size and cost.

3. Proposed LQRC and LQRIC for LC-HAPF

3.1. Linear Quadratic Regulator Control (LQRC) for LC-HAPF

3.1.1. State-Space Model for LQRC-Controlled LC-HAPF

Based on (4), the small-signal state-space model is proposed for the LQRC, and the detail derivation is given as follows. In order to extract the reactive power and harmonics from the three-phase power grid, the d-q-0 synchronous rotating coordinate system is applied. Through the a-b-c to d-q-0 conversion, the active power, harmonics, reactive power and three-phase unbalance convert into d-axis DC component, d-axis AC component, q-axis and 0-axis, respectively. The a-b-c to d-q-0 conversion is achieved by the Park transform matrix H given in (5). After the reference current calculation in the d-q-0 coordinate, the calculated reference compensation current will convert back to an a-b-c coordinate by inverse Park transform H⁻¹.

$$\mathrm{H}=\frac{2}{3}\left[\begin{array}{ccc}\cos ({\omega }_{1}t)& \cos ({\omega }_{1}t-120°)& \cos ({\omega }_{1}t+120°)\\ -\sin ({\omega }_{1}t)& -\sin ({\omega }_{1}t-120°)& -\sin ({\omega }_{1}t+120°)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]$$

(5)

where ω₁t is the fundamental phase angle and is obtained by phase locked loop (PLL). With the help of (5), i_cx, v_invx, v_x and v_cx in the d-q-0 coordinate can be obtained. They are shown as:

$$\{\begin{array}{l}{\left[\begin{array}{ccc}{i}_{cd}& i_{cq}& i_{c0}\end{array}\right]}^T=\mathrm{H}{\left[\begin{array}{ccc}{i}_{ca}& i_{cb}& i_{cc}\end{array}\right]}^T\\ {\left[\begin{array}{ccc}{v}_{invd}& v_{invq}& v_{inv0}\end{array}\right]}^T=\mathrm{H}{\left[\begin{array}{ccc}{v}_{inva}& v_{invb}& v_{invc}\end{array}\right]}^T\\ {\left[\begin{array}{ccc}{v}_d& v_q& v_0\end{array}\right]}^T=\mathrm{H}{\left[\begin{array}{ccc}{v}_a& v_b& v_c\end{array}\right]}^T\\ {\left[\begin{array}{ccc}{v}_{cd}& v_{cq}& v_{c0}\end{array}\right]}^T=\mathrm{H}{\left[\begin{array}{ccc}{v}_{ca}& v_{cb}& v_{cc}\end{array}\right]}^T\end{array}$$

(6)

From (1), (2), (5) and (6), the state differential equations for the LC-HAPF in d-q-0 coordinate are given by (7), and the outputs of the LC-HAPF in the d-q-0 coordinate are given by (8).

$$\{\begin{array}{l}\frac{d{i}_{cd}}{dt}=-\frac{R_{cx}}{L_{cx}}{i}_{cd}+{\omega }_1{i}_{cq}+\frac{1}{L_{cx}}{v}_{invd}-\frac{1}{L_{cx}}(v_d-v_{cd})\\ \frac{d{i}_{cq}}{dt}=-{\omega }_1{i}_{cd}-\frac{R_{cx}}{L_{cx}}{i}_{cq}+\frac{1}{L_{cx}}{v}_{invq}-\frac{1}{L_{cx}}(v_q-v_{cq})\\ \frac{d{i}_{c0}}{dt}=-\frac{R_{cx}}{L_{cx}}{i}_{c0}+\frac{1}{L_{cx}}{v}_{inv0}-\frac{1}{L_{cx}}(v_0-v_{c0})\end{array}$$

(7)

$$\{\begin{array}{l}{y}_d=i_{cd}\\ y_q=i_{cd}\\ y_0=i_{c0}\end{array}$$

(8)

For the steady state system in (7), overlaying perturbations ${\tilde{i}}_{ch}$, ${\tilde{v}}_{invh}$, ${\tilde{v}}_h$ and ${\tilde{v}}_{ch}$ are given by (9), where the subscript h indicates the phase d, q and 0. [16] and the ‘~’ denotes the perturbation value.

$$\{\begin{array}{l}{i}_{ch}=I_{ch}+{\tilde{i}}_{ch},\\ v_{invh}=V_{invh}+{\tilde{v}}_{invh},\\ v_h=V_h+{\tilde{v}}_h,\\ v_{ch}=V_{ch}+{\tilde{v}}_{ch}.\end{array}$$

(9)

where I_ch, V_invh, V_h and V_ch are the steady state values of compensation current, inverter voltage, PCC voltage and compensation capacitor voltage.

Based on (7)–(9), the small-signal state-space model of the LC-HAPF with LQRC-controlled is then set up and shown in (10).

$$\{\begin{array}{l}\frac{d}{dt}\tilde{\mathrm{e}}=\mathrm{A}\tilde{\mathrm{e}}+\mathrm{B}\tilde{\mathrm{u}}+\mathrm{E}\tilde{\mathrm{z}}\\ \tilde{\mathrm{y}}=\mathrm{C}\tilde{\mathrm{e}}+\mathrm{D}\tilde{\mathrm{u}}\end{array}$$

(10)

where $\tilde{\mathrm{e}}$, $\tilde{\mathrm{u}}$, $\tilde{\mathrm{z}}$ and $\tilde{\mathrm{y}}$ are expressed as $\tilde{\mathrm{e}}={\left[\begin{array}{ccc}{\tilde{i}}_{cd}& {\tilde{i}}_{cq}& {\tilde{i}}_{c0}\end{array}\right]}^T$, $\tilde{\mathrm{u}}={\left[\begin{array}{ccc}{\tilde{v}}_{invd}& {\tilde{v}}_{invq}& {\tilde{v}}_{inv0}\end{array}\right]}^T$, $\tilde{\mathrm{z}}={\left[\begin{array}{ccc}{\tilde{v}}_d-{\tilde{v}}_{cd}& {\tilde{v}}_q-{\tilde{v}}_{cq}& {\tilde{v}}_0-{\tilde{v}}_{c0}\end{array}\right]}^T$ and $\tilde{\mathrm{y}}={\left[\begin{array}{ccc}{\tilde{i}}_{cd}& {\tilde{i}}_{cq}& {\tilde{i}}_{c0}\end{array}\right]}^T$. The matrices A, B, C, D and E are expressed as:

$$\begin{array}{l}\mathrm{A}=\left[\begin{array}{ccc}-\frac{R_{cx}}{L_{cx}}& {\omega }_1& 0\\ -{\omega }_1& -\frac{R_{cx}}{L_{cx}}& 0\\ 0& 0& -\frac{R_{cx}}{L_{cx}}\end{array}\right], \mathrm{B}=\left[\begin{array}{ccc}\frac{1}{L_{cx}}& 0& 0\\ 0& \frac{1}{L_{cx}}& 0\\ 0& 0& \frac{1}{L_{cx}}\end{array}\right],\\ \mathrm{C}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right], \mathrm{D}=0, \mathrm{E}=-\left[\begin{array}{ccc}\frac{1}{L_{cx}}& 0& 0\\ 0& \frac{1}{L_{cx}}& 0\\ 0& 0& \frac{1}{L_{cx}}\end{array}\right].\end{array}$$

So far, the small-signal state-space model of the LQRC-controlled LC-HAPF is established for the derivation process of the LQRC for the LC-HAPF. And it applies in the following section.

3.1.2. LQRC for LC-HAPF

The objective of optimal control is to find a control system under the given constraint conditions so as to minimize the cost function, J. LQRC, which is an optimal control, has good tracking control ability, and calculates the gain by the cost function, weighting matrices and state-space model. Generally, LQRC takes the minimum value of the cost function (11) in time interval [t₀, t_f] [17], where t₀ and t_f represent the initial time and final time, respectively.

$$J={\displaystyle \underset{t_0}{\overset{t_f}{\int }}\left({\mathrm{e}}^{\mathrm{T}}\mathrm{Qe}+{\mathrm{u}}^{\mathrm{T}}\mathrm{Ru}\right)}dt$$

(11)

Q is a state weighting matrix, which is defined as a diagonal semi-positive matrix ($\mathrm{Q}\ge 0$), and measures the importance of each phase state variable for the LQRC. R is an input weighting matrix, which is diagonal positive definite ($\mathrm{R}>0$) and measures the importance level of each phase control variable. They are given by (12) and (13).

$$\mathrm{Q}=\left[\begin{array}{ccc}{q}_{q1}& 0& 0\\ 0& q_{q2}& 0\\ 0& 0& q_{q3}\end{array}\right]$$

(12)

$$\mathrm{R}=\left[\begin{array}{ccc}{r}_{q1}& 0& 0\\ 0& r_{q2}& 0\\ 0& 0& r_{q3}\end{array}\right]$$

(13)

when e and u indicate the error and input, q_qi (i = 1, 2, 3, …) represents the weighting value of corresponding e, and r_qi represents the weighting value of corresponding u. The larger value of q_qi means take more effect on corresponding state variable in certain phase, so as r_qi.

The physical meaning of minimizing the cost function (11) is to minimize the tracking error and control cost (DC-link voltage of the inverter) by minimizing the corresponding terms ${\mathrm{e}}^{\mathrm{T}}\mathrm{Qe}$ and ${\mathrm{u}}^{\mathrm{T}}\mathrm{Ru}$.

To solve the optimal control problem for this dynamical system, the Hamiltonian function F with Lagrange multipliers λ in (14) is constructed.

$$F\left[\mathrm{e},\mathrm{u},\mathsf{\lambda },t\right]=\frac{1}{2}\left[{\mathrm{e}}^{\mathrm{T}}\mathrm{Qe}+{\mathrm{u}}^{\mathrm{T}}\mathrm{Ru}\right]+{\mathsf{\lambda }}^{\mathrm{T}}\left[\mathrm{Ae}+\mathrm{Bu}\right]$$

(14)

Then, (15) is used to calculate the minimum value of F, i.e., the minimum value of the cost function, J.

$$\frac{\partial F}{\partial \mathrm{u}}=\mathrm{Ru}+{\mathrm{B}}^{\mathrm{T}}\mathsf{\lambda }=0$$

(15)

As R is positive and symmetrical, by (15), the optimal control variable solution u* is calculated by:

$${\mathrm{u}}^{*}=-{\mathrm{R}}^{-1}{\mathrm{B}}^{\mathrm{T}}\mathsf{\lambda }$$

(16)

In order to realize feedback control, the positive-definite transformation matrix, P, is introduced to evaluate the λ.

$$\mathsf{\lambda }=\mathrm{P}\cdot \mathrm{e}$$

(17)

and the matrix P is the solution of Algebraic Riccati equation (ARE):

$$\mathrm{PA}+{\mathrm{A}}^{\mathrm{T}}\mathrm{P}-{\mathrm{PBR}}^{-1}{\mathrm{B}}^{\mathrm{T}}\mathrm{P}+\mathrm{Q}=0$$

(18)

From (16) and (17), we have:

$${\mathrm{u}}^{*}=-{\mathrm{R}}^{-1}{\mathrm{B}}^{\mathrm{T}}\mathrm{P}\cdot \mathrm{e}=\mathrm{K}\cdot \mathrm{e}$$

(19)

$$\mathrm{K}={\mathrm{R}}^{-1}{\mathrm{B}}^{\mathrm{T}}\mathrm{P}$$

(20)

where K is the gain matrix for LQR optimal control. Every combination of Q and R matrices can obtain a unique solution for K, which can satisfy the minimum cost of (11).

From (19) and (20), the controlled system is expressed as:

$$\frac{d}{dt}\mathrm{e}=\mathrm{Ae}+\mathrm{Bu}=\left(\mathrm{A}-\mathrm{BK}\right)\mathrm{e}$$

(21)

and the design of Q and R matrices is based on Bryson’s rule [30], which provides the reasonable weighting elements q_qi and r_qi that are shown in below:

$$\{\begin{array}{l}{q}_{qi}=\frac{1}{Maximumacceptablevalueof\left[e_i^2\right]}\\ r_{qi}=\frac{1}{Maximumacceptablevalueof\left[u_i^2\right]}\end{array}$$

(22)

The overall design process of LQRC is shown as follows:

• Select Q and R matrices based on Bryson’s rule;
• Substitute the A, B, Q and R matrices to (18) and obtain the P matrix;
• Using the R, B and P matrices to obtain the optimal control gain K matrix.

3.2. LQRC with Integral Action (LQRIC) for LC-HAPF

The LQRIC adds the integral term of each state variable on the basis of LQRC, which aims to reduce the steady state error of the system. Thus, for the small-signal state-space model of LQRIC-controlled LC-HAPF, the state variable $\tilde{\mathrm{e}}$ is upgraded and shown in (23), whereas the control variable $\tilde{\mathrm{u}}$, disturbance variable $\tilde{\mathrm{z}}$, and output variable $\tilde{\mathrm{y}}$ of the LQRIC remain the same as the LQRC.

$$\tilde{\mathrm{e}}={\left[\begin{array}{cccccc}{\tilde{i}}_{cd}& {\tilde{i}}_{cq}& {\tilde{i}}_{c0}& {\displaystyle \int {\tilde{i}}_{cd}}& {\displaystyle \int {\tilde{i}}_{cq}}& {\displaystyle \int {\tilde{i}}_{c0}}\end{array}\right]}^T$$

(23)

The matrices A, B, C, D and E for the LQRIC-controlled LC-HAPF are modified accordingly and expressed as:

$$\begin{array}{r}\mathrm{A}=\left[\begin{array}{cccccc}-\frac{R_{cx}}{L_{cx}}& \omega & 0& 0& 0& 0\\ -\omega & -\frac{R_{cx}}{L_{cx}}& 0& 0& 0& 0\\ 0& 0& -\frac{R_{cx}}{L_{cx}}& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\end{array}\right], \mathrm{B}=\left[\begin{array}{ccc}\frac{1}{L_{cx}}& 0& 0\\ 0& \frac{1}{L_{cx}}& 0\\ 0& 0& \frac{1}{L_{cx}}\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\\ \mathrm{C}=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\end{array}\right], \mathrm{D}=0, \mathrm{E}=-\left[\begin{array}{ccc}\frac{1}{L_{cx}}& 0& 0\\ 0& \frac{1}{L_{cx}}& 0\\ 0& 0& \frac{1}{L_{cx}}\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right].\end{array}$$

For LQRIC, the R matrix remains the same as LQRC, whereas the Q matrix has three additional integral weighting elements, q_q4, q_q5 and q_q6, which are shown below:

$$\mathrm{Q}=\left[\begin{array}{cccccc}{q}_q{}_1& 0& 0& 0& 0& 0\\ 0& q_q{}_2& 0& 0& 0& 0\\ 0& 0& q_q{}_3& 0& 0& 0\\ 0& 0& 0& q_q{}_4& 0& 0\\ 0& 0& 0& 0& q_q{}_5& 0\\ 0& 0& 0& 0& 0& q_q{}_6\end{array}\right]$$

(24)

For the design of the Q matrix for LQRIC, the integral weighting elements are designed larger than the corresponding proportional weighting elements q_q1, q_q2 and q_q3, i.e., $q_{q4}\ge q_{q1}, q_{q5}\ge q_{q2} \mathrm{a}\mathrm{n}\mathrm{d} q_{q6}\ge q_{q3}$, in order to achieve a better steady-state performance compared with the LQRC. Finally, based on the updated variable $\tilde{\mathrm{e}}$ and matrices A, B, C, D, E, Q and R, the optimal control gain K matrix of the LQRIC can be obtained by the design process of LQRC in Section 3.1.2, and therefore the controlled system in (21) of LQRIC can be acquired.

4. Overall Control Strategy of LC-HAPF

Figure 3 shows the overall control block diagram of the LQRC and LQRIC for the LC-HAPF. Firstly, both i_Lx and i_cx are transformed into d-q-0 coordinates by Park transformation, where the ω₁t is obtained by the PLL. The reference compensation current in the d-axis $i_{cd}^{*}$ is derived from the high pass filter (HPF). Then, the compensation current error ε_h is obtained by ε_h = i_ch − i_ch^*. After that, the ε_h can be transformed into v_invh^* by the LQRC or LQRIC, and the v_invh^* is converted back into the a-b-c coordinate, which is v_invx^*. It will be applied into the PWM so as to generate the inverter trigger signals T_1x and T_2x to control the IGBTs in the VSI of LC-HAPF. Finally, the VSI outputs v_invx to the PPF branch in Figure 1, in which i_cx injects into the power grid and compensates the reactive power and harmonics current.

In this paper, a comparative study of the compensation for LC-HAPF with different controllers will be demonstrated in order to verify the effectiveness of the LQRCI for LC-HAPF. The HCC and PCC are applied to the LC-HAPF for the comparative study, and their control block diagrams and parameter designs will be mentioned in the following sections.

4.1. Hysteresis Current Control (HCC)

The control block diagram of HCC [14] is shown in Figure 4a. The HCC is derived by:

$$v_{invx}=\{\begin{array}{c}\frac{V_{dc}}{2}, \mathrm{f}\mathrm{o}\mathrm{r} {\epsilon }_x\ge B;\\ -\frac{V_{dc}}{2}, \mathrm{f}\mathrm{o}\mathrm{r} {\epsilon }_x\le B.\end{array}$$

(25)

The design of the hysteresis band B needs to satisfy (26).

$$B=\frac{V_{dc}}{8L_{cx}{f}_{sw}}$$

(26)

where f_sw is the switching frequency.

4.2. Proportional Current Control (PCC)

The control block diagram of PCC [15] is shown in Figure 4b. The PCC is expressed by:

$$v_{invx}^{*}=K_p{\epsilon }_x$$

(27)

where K_p is the proportional gain and the K_p can be designed by:

$$0<K_p\le \frac{8L_{cx}}{3T_s}$$

(28)

where T_s is the sampling time.

5. Simulation Result

The system parameters and control parameters of different controllers for simulation are shown in

Table 1. System parameters and control parameters of different controllers for simulation.

System Parameters	Values	Controller Parameters	Values
vsx, f	110 Vrms, 50 Hz	H (HCC)	0.156
Ls	0.5 mH	Kp (PCC)	250
LLx, CLx, RLx	35 mH, 400 μF, 43 Ω	qq1, qq2, qq3, rq1, rq2, rq3 (LQRC)	350, 310, 370, 0.01, 0.01, 0.01
Lcx, Ccx, Rcx	8 mH, 50 μF, 0.03 Ω	qq1, qq2, qq3, qq4, qq5, qq6, rq1, rq2, rq3 (LQRIC)	260, 240, 290, 830, 820, 450, 0.01, 0.01, 0.01
Vdc	100 V
fs	10 kHz

. MATLAB/Simulink is used to simulate the compensation results of the different current controllers for the three-phase four-wire LC-HAPF. The simulation results before and after HCC, PCC, LQRC and LQRIC-controlled LC-HAPF under 50 V and 40 V DC-link voltage conditions are shown in Figure 5 and Figure 6, and the values are summarized in

Table 2. Simulation results before and after LC-HAPF compensation under 50V DC-link voltage.

	Before Compensation	After Compensation
		HCC	PCC	LQRC	LQRIC
THDisa (%)	33.7	11.3	8.3	7.4	6.2
THDisb (%)	33.7	10.9	8.7	7.9	6.8
THDisc (%)	33.7	11.3	8.7	8.1	6.8
PF	0.76	0.99	1.00	1.00	1.00
QTotal (var)	615.1	4.2	3.8	2.8	2.1
Total SW loss (W)	/	25.0	26.3	26.6	27.0
isa (Arms)	3.28	2.50	2.48	2.47	2.46
isb (Arms)	3.28	2.50	2.47	2.47	2.46
isc (Arms)	3.28	2.49	2.48	2.47	2.46
isn (Arms)	2.97	0.49	0.44	0.41	0.38

and

Table 3. Simulation results before and after the LC-HAPF compensation under 40V DC-link voltage.

	Before Compensation	After Compensation
		HCC	PCC	LQRC	LQRIC
THDisa (%)	33.7	15.4	14.4	8.2	6.1
THDisb (%)	33.7	15.6	15.0	7.9	6.3
THDisc (%)	33.7	15.9	14.3	8.0	7.1
PF	0.76	0.99	0.99	1.00	1.00
QTotal (var)	615.1	10.0	9.7	3.6	2.9
Total SW loss (W)	/	12.3	13.8	15.9	15.9
isa (Arms)	3.28	2.49	2.46	2.46	2.46
isb (Arms)	3.28	2.49	2.45	2.46	2.46
isc (Arms)	3.28	2.48	2.45	2.46	2.46
isn (Arms)	2.97	0.79	0.77	0.41	0.36

.

5.1. Hysteresis Current Control (HCC)

After compensation, the simulation results of HCC under 50V DC-link voltage conditions are shown in Figure 5a and Table 2. The harmonic currents are compensated, and the total source current harmonic distortion (THD_isx) is reduced from 33.7% to about 11.3%. Also, the compensated i_sn is 0.49 Arms and the compensated i_sx is around 2.50 Arms. Additionally, the three-phase power factors (PFs) are the same and increase to 0.99, and the Q_Total reduces to 4.2 var. The total switching loss (total SW loss) of HCC is 25.0 W.

The simulation results of HCC under 40V DC-link voltage conditions are shown in Figure 6a and Table 3. The THD_isx reduces to about 15.9%. Also, the compensated i_sn is 0.79 Arms and the compensated i_sx is around 2.49 Arms. Additionally, the three-phase PFs are the same and increase to 0.99, and the Q_Total reduces to 10.0 var. The total SW loss of HCC is 12.3 W.

5.2. Proportional Current Control (PCC)

The simulation results of the LC-HAPF with PCC under 50V DC-link voltage conditions are shown in Figure 5b and Table 2. The THD_isx reduces to about 8.7% after compensation. The compensated i_sn is 0.44 Arms and the compensated i_sx is around 2.48 Arms. Moreover, the three-phase PFs are equal to 1, and the Q_Total reduces to 3.8 var. The total SW loss of PCC is 26.3 W.

The simulation results of the LC-HAPF with PCC under 40V DC-link voltage conditions are shown in Figure 6b and Table 3. The THD_isx reduces to about 15.0% after compensation. The compensated i_sn is 0.77 Arms and the compensated i_sx is around 2.45 Arms. Moreover, the three-phase PFs increase to 0.99, and the Q_Total reduces to 9.7 var. The total SW loss of PCC is 13.8 W.

5.3. Linear Quadratic Regulator Control (LQRC)

Figure 5c and Table 2 show the simulation results of the LC-HAPF with LQRC under 50V DC-link voltage conditions. After compensation, the THD_isx reduces to about 8.1%. The compensated i_sn is 0.41 Arms and the compensated i_sx is around 2.47 Arms. The three-phase PFs are equal to 1, and the Q_Total reduces to 2.8 var. The total SW loss of LQRC is 26.6 W.

Figure 6c and Table 3 show the simulation results of the LC-HAPF with LQRC under 40V DC-link voltage conditions. After compensation, the THD_isx reduces to about 8.2%. The compensated i_sn is 0.41 Arms and the compensated i_sx is around 2.46 Arms. The three-phase PFs are equal to 1, and the Q_Total reduces to 3.6 var. The total SW loss of LQRC is 15.9 W.

5.4. Proposed LQR Control with Integral Action Control (LQRIC)

The simulation results of the LC-HAPF with proposed LQRIC under 50V DC-link voltage conditions are shown in Figure 5d and Table 2. After compensation, the THD_isx reduces to about 6.8%. The compensated i_sn is 0.38 Arms and the compensated i_sx is around 2.46 Arms. The three-phase PFs are equal to 1, and the Q_Total reduces to 2.1 var. The total SW loss of LQRC is 27.0 W.

The simulation results of the LC-HAPF with proposed LQRIC under 40V DC-link voltage conditions are shown in Figure 6d and Table 3. After compensation, the THD_isx reduces to about 7.1%. The compensated i_sn is 0.36 Arms and the compensated i_sx is around 2.46 Arms. The three-phase PFs are equal to 1, and the Q_Total reduces to 2.9 var. Hence, the proposed LQRIC can achieve the smallest steady-state error. The total SW loss of LQRC is 15.9 W.

6. Experimental Result

To verify the feasibility of proposed LQRIC, a 110 V–5 kVA three-phase four-wire LC-HAPF experimental prototype was constructed and is shown in Figure 7. A digital signal processor TMS320F2812 is applied and works in a 10 kHz sampling frequency. The Mitsubishi IGBT intelligent power module PM300DSA60 is used as the power switch in the VSI.

This paper takes IEEE Standard 519-2014 [31] as a reference. At small experimental rated 110 V–5 kVA prototypes and I_SC/I_L in the range of 100 < 1000 scale, the acceptable total demand distortion (TDD) is less than or equal to 15%. Considering the worst case, the rated current is equal to the fundamental load current and results in THD = TDD ≤ 15%. Therefore, THD ≤ 15% is used as a criterion to evaluate the LC-HAPF current harmonic compensation performance.

The experimental results before and after the LC-HAPF compensation with LQRC and LQRIC under 50V and 40V DC-link voltage conditions are shown in Figure 8 and Figure 9 and

Table 4. Experimental results before and after the LC-HAPF compensation with LQRC and LQRIC.

	Before Compensation	50V DC-Link Voltage		40V DC-Link Voltage
		LQRC	LQRIC	LQRC	LQRIC
THDisa (%)	26.2	10.1	8.8	10.7	9.2
THDisb (%)	26.7	9.6	8.1	10.7	8.8
THDisc (%)	26.2	10.0	8.8	10.6	9.1
PF	0.80	0.99	0.99	0.99	0.99
QTotal (var)	580	20	20	20	20
Total SW loss (W)	/	29.6	30.2	19.5	20.3
isa (Arms)	3.4	2.9	3.0	2.9	2.9
isb (Arms)	3.4	2.9	2.9	2.9	2.9
isc (Arms)	3.4	2.9	2.9	2.9	2.9
isn (Arms)	2.4	0.9	0.8	1.0	0.9

.

6.1. Linear Quadratic Regulator Control (LQRC)

After compensation, the experimental results of LQRC under 50V DC-link voltage conditions are shown in Figure 8a and Figure 9a and Table 4. The THD_isx reduces to about 10.1%. Also, the compensated i_sn is 0.9 Arms and the compensated i_sx is 2.9 Arms. Additionally, the three-phase PFs are the same and increase to 0.99, and the Q_Total reduces to 20 var. The total SW loss is 29.6 W.

The experimental results of LQRC under 40V DC-link voltage conditions are shown in Figure 8b and Figure 9b and Table 4. The harmonic currents are compensated, and the THD_isx reduces to about 10.7%. Also, the compensated i_sn is 1.0 Arms and the compensated i_sx is 2.9 Arms. Moreover, the three-phase PFs are equal and increase to 0.99, and the Q_Total reduces to 20 var. The total SW loss is 19.5 W.

6.2. Proposed LQR Control with Integral Action Control (LQRIC)

The experimental results of LQRIC under 50V DC-link voltage conditions are shown in Figure 8c and Figure 9c and Table 4. The THD_isx reduces to about 8.8% after compensation. The compensated i_sn is 0.8 Arms and the compensated i_sx is around 2.9 Arms. Additionally, the three-phase PFs are the same and increase to 0.99, and the Q_Total reduces to 20 var. The total SW loss is 30.2 W.

The experimental results of LQRIC under 40V DC-link voltage conditions are shown in Figure 8d and Figure 9d and Table 4. The THD_isx reduces to about 9.2% after compensation. The compensated i_sn is 0.9 Arms and the compensated i_sx is around 2.9 Arms. Moreover, the three-phase PFs are equal and increase to 0.99, and the Q_Total reduces to 20 var. The total SW loss is 20.3 W.

7. Discussion

7.1. Simulation Results Comparison of Different Controllers

From the simulation results shown in Figure 5 and Table 2, all controllers can compensate the reactive power and current harmonics. Comparing four controllers, HCC shows the worst THDi_sx, the largest neutral current i_sn and the largest reactive power after compensation, the current after HCC compensation still exists as a large current ripple. The PCC shows a lower THDi_sx, i_sn and Q_Total than HCC with the fixed switching frequency, whereas the K_p gain of PCC is obtained by trial and error. LQRC shows a better harmonic, reactive power and neutral current compensation capability compared with PCC, as its control gain matrix K is optimized using the LQR method. LQRIC has the best harmonic and reactive power compensation capability with the lowest THD_isx, Q_Total and i_sn because an integral action is added to perform the error correction and the control gain K is also optimized by the LQR method.

7.2. Optimal Control of LQRC and LQRIC

From the simulation results shown in Figure 6 and Table 3, when the supplied DC-link voltage is reduced from 50V to 40V, the compensation performances of HCC and PCC become worse. A large steady-state error and reactive power exists after the HCC- and PCC-controlled LC-HAPF compensation, whereas the LQRC maintains a good harmonic and reactive power compensation performance. The LQRIC still shows the lowest THD_isx, i_sn and Q_Total, which verifies the optimal control of the proposed LQRIC.

7.3. Feasibility of LQRC and LQRIC Controllers

The experimental results in Figure 8, Figure 9 and Table 4 verify the feasibility of LQRC and LQRIC for LC-HAPF under 50V and 40V DC-link voltage. Followed by the IEEE standard [31], the LQRC- and LQRIC-controlled LC-HAPF can compensate the THD_isx to less than 15%, while the reactive power and neutral current are small after compensation. These results show that both LQRC and LQRIC can achieve optimal control and can compensate harmonics and reactive power even in low DC-link voltage conditions. The proposed LQRIC also shows a better steady-state and reactive power compensation performance compared with the LQRC.

8. Conclusions

In this paper, a LQRIC for the three-phase four-wire LC-HAPF is proposed and designed to ensure a good steady-state performance even under low DC-link voltage. The state-space model in the d-q-0 coordinate of the LQRIC-controlled LC-HAPF is established and helps to obtain the reference current. The detailed design process of the weighting matrices, Q and R, of the LQRIC for the LC-HAPF based on Bryson’s rule is also given in this paper. Then, the control gain K matrix is calculated through the LQR method by using MATLAB. Finally, the obtained K matrix is applied and the feasibility of LQRC and LQRIC for the LC-HAPF are verified by the simulation and experimental results. The results show that LQRC- and LQRIC-controlled LC-HAPF can achieve a good compensation performance even under low DC-link voltage. Moreover, the proposed LQRIC shows superior steady-state performance compared with the harmonic compensation, neutral current compensation and reactive power compensation with HCC, PCC and LQRC.