A Novel Multi-vector Model Predictive Current Control of Three-Phase Active Power Filter

The MPC control method relies on the model of the APF system to predictive how the possible control actions would affect its response. So, the action that is expected to minimize a certain cost function is applied and the process is sequentially repeated. To obtain good control performance when this method is applied to APF, the appropriate model of APF is needed [18, 19].

Figure 1 shows the system main structure of a parallel APF, where e_a, e_b, e_c are grid voltage, i_ca, i_cb, i_cc are the output compensation current of APF, i_La, i_Lb, i_Lc are load current, where L and R are the filter inductance and equivalent resistance, C is DC side capacitance, which stores energy for bidirectional flow of APF and grid energy, for U_dc of the DC voltage hold stability.

1.png

Figure 1. Three-phase parallel APF structure diagram

Three groups of switches $S_{x} \in\left\{S_{a}, S_{b}, S_{c}\right\}$ are defined as two states: on and off. To prevent simultaneous conduction failure, dead zone protection is added to the output switch. These switches are defined as Eq. (1).

${{S}_{a}}=\left\{ \begin{align}  & 1;\quad \,\text{if}\ {{S}_{1}}\,\text{is}\ \operatorname{ON}\ \text{and}\ {{S}_{4}}\ \text{is}\ \operatorname{OFF} \\  & \text{0};\quad \text{if}\ {{S}_{1}}\,\text{is}\ \operatorname{OFF}\ \text{and}\ {{S}_{4}}\ \text{is}\ \operatorname{ON} \\ \end{align} \right.$

${{S}_{b}}=\left\{ \begin{align}  & 1;\quad \,\text{if}\ {{S}_{2}}\,\text{is}\ \operatorname{ON}\ \text{and}\ {{S}_{5}}\ \text{is}\ \operatorname{OFF} \\  & \text{0};\quad \text{if}\ {{S}_{2}}\,\text{is}\ \operatorname{OFF}\ \text{and}\ {{S}_{5}}\ \text{is}\ \operatorname{ON} \\ \end{align} \right.$             (1)

${{S}_{c}}=\left\{ \begin{align}  & 1;\quad \,\text{if}\ {{S}_{3}}\,\text{is}\ \operatorname{ON}\ \text{and}\ {{S}_{6}}\ \text{is}\ \operatorname{OFF} \\  & \text{0};\quad \text{if}\ {{S}_{3}}\,\text{is}\ \operatorname{OFF}\ \text{and}\ {{S}_{6}}\ \text{is}\ \operatorname{ON} \\ \end{align} \right.$

Therefore, the overall APF can be in eight (2³) possible switch configurations, and the point N are calculated by multiplying the DC link voltage with the state of the respective leg:

${{v}_{aN}}={{S}_{a}}\cdot {{v}_{dc}}$

${{v}_{bN}}={{S}_{b}}\cdot {{v}_{dc}}$      (2)

${{v}_{cN}}={{S}_{c}}\cdot {{v}_{dc}}$

It is necessary to subtract the common-mode voltage v_nN needs from (2) to obtain the effective voltage applied to each phase (from points a, b and c to n). The common mode voltage can be simply calculated by considering Kirchhoff's voltage law:

${{v}_{nN}}=\frac{{{v}_{aN}}+{{v}_{bN}}+{{v}_{cN}}}{3}$        (3)

Therefore, the effective phase voltage can be obtained:

${{v}_{an}}={{v}_{aN}}-{{v}_{nN}}$

${{v}_{bn}}={{v}_{bN}}-{{v}_{nN}}$             (4)

${{v}_{cn}}={{v}_{cN}}-{{v}_{nN}}$

Assuming three-phase symmetry, according to the delay of Eq. (5), the output of APF can be defined as Eq. (6), where v_aN, v_bN, and v_cN are the voltages of each phase relative to N point.

$\text{a}={{e}^{j2\pi /3}}=-\frac{1}{2}+j\frac{\sqrt{3}}{2}$              (5)

$\text{v}=\frac{2}{3}({{v}_{aN}}+\text{a}{{v}_{bN}}+{{\text{a}}^{2}}{{v}_{cN}})$        (6)

The vector composed of S_x in different switching states of three-phase two-level APF, by applying the complex Clark transformation (1) to (8) for eight possible switch configurations, the input vectors can be shown in Figure 2.

2.png

Figure 2. Three-phase two-level parallel APF switch vector in the complex α-β frame

2.1 Discrete mathematical model of three-phase APF

In the three-phase static coordinate system, assuming the three-phase symmetry, according to Kirchhoff's current law, the dynamic expression of Eq. (7) can be obtained, where u_ca, u_cb, and u_cc are the output voltage of APF [20].

$\left\{ \begin{align}  & L\frac{\text{d}{{i}_{a}}}{\text{d}t}={{e}_{a}}-R{{i}_{a}}-{{u}_{c}}_{a} \\  & L\frac{\text{d}{{i}_{b}}}{\text{d}t}={{e}_{b}}-R{{i}_{b}}-{{u}_{c}}_{b} \\  & L\frac{\text{d}{{i}_{c}}}{\text{d}t}={{e}_{c}}-R{{i}_{c}}-{{u}_{c}}_{c} \\ \end{align} \right.$         (7)

According to Eq. (7), the state variable is set as X=[i_a;i_b;i_c] and the input is U = [e_a-u_ca; e_b-u_cb; e_c-u_cc], and the state space model of three-phase two-level APF is established.

$\mathbf{\dot{X}}={{\mathbf{A}}_{3\times 3}}\mathbf{X}+{{\mathbf{B}}_{3\times 3}}\mathbf{U}$        (8)

A and B are state matrix and input matrix, which are determined by model parameters of APF. Among,

$\mathbf{A}=\{{{(-\frac{R}{L})}_{ii}}\},\mathbf{B}=\{{{(\frac{1}{L})}_{ii}}\},i=1,2,3$                 (9)

By transforming it into d-q coordinate system and discretizing it, we can get Eq. (10):

$\mathbf{\tilde{X}}(k+1)={{\mathbf{\tilde{\Psi }}}_{2\times 2}}\mathbf{\tilde{X}}(k)+{{\mathbf{\tilde{\Theta }}}_{2\times 2}}\mathbf{\tilde{U}}$              (10)

where,

$\mathrm{\tilde{X}}={{C}_{dq}}{{C}_{32}}\mathbf{X};\mathrm{\tilde{U}=}{{C}_{dq}}{{C}_{32}}\mathbf{U}$      (11)

$\mathbf{\tilde{\Psi }}={{\operatorname{e}}^{\mathbf{\Psi }{{T}_{s}}}};\mathbf{\tilde{\Theta }}=({{\operatorname{e}}^{\mathbf{\Psi }{{T}_{s}}}}-\mathbf{I}){{\mathbf{\Psi }}^{-1}}\mathbf{\Theta }$         (12)

$\begin{align}  & \mathbf{\Psi }=\left[ \begin{matrix}   -\frac{R}{L} & 0  \\   0 & -\frac{R}{L}  \\\end{matrix} \right],\mathbf{\Theta }=\left[ \begin{matrix}   \frac{1}{L} & {}  \\   {} & \frac{1}{L}  \\\end{matrix} \right] \\  & {{C}_{dq}}=\sqrt{\frac{2}{3}}\left[ \begin{matrix}   \sin \omega t & -\cos \omega t  \\   -\cos \omega t & -\sin \omega t  \\\end{matrix} \right] \\ \end{align}$

${{C}_{32}}=\sqrt{\frac{2}{3}}\left[ \begin{matrix}   1 & -\frac{1}{2} & \frac{1}{2}  \\   0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2}  \\\end{matrix} \right]$             (13)

Eq. (10) is the discrete mathematical model in d-q coordinate system, in (12), the T_S is the sampling period of the system, in (13), the ꞷ is the phase angle detected by PLL.

2.2 Calculation of reference current

3.png

Figure 3. i_p-i_q method for harmonic detection of the reference current

When APF is used to compensate harmonic and reactive power, it must be detected first, including FFT, FBD, PQ method, and so on. At present, the most widely used i_p-i_q detection method proposed by H. Akagi is that the fundamental wave of load current is obtained by filtering, and the harmonic content of load current can be obtained by subtracting the load current [21]. The DC side capacitor voltage of APF is controlled by PI, and the compensation part is injected to keep the energy interaction between AC measurement and DC side and stabilize it at the reference value [22, 23]. It has better dynamic performance and tracking performance, and the specific principle is shown in Figure 3.

The detected three-phase harmonics i_ah, i_bh and i_ch are taken as reference values, it is X_ref = [i_ah;i_bh;i_ch]. In the harmonic detection link, there is a certain time delay from current detection to harmonic calculation, and it cannot be used as the reference value of the predicted value at the next moment. The reference value can be predicted by Lagrange interpolation, and Eq. (14) can be obtained. The error of second-order interpolation and third-order interpolation is small, in order to reduce the amount of calculation, n = 2 is adopted in this paper.

${{\mathbf{\tilde{X}}}_{ref}}(k+1)=\sum\limits_{i=0}^{n}{{{(-1)}^{n-i}}\frac{(n+1)!}{i!(n+1-i)!}\cdot {{\mathbf{X}}_{ref}}(k+i-n\,)\ }$         (14)

Multi-vector MPCC is a control method based on the optimal duty cycle model [15], the specific principle is shown in Figure 5. The traditional multi-vector MPCC control obtains the optimal vector V_opt1 through the calculation of first loop the vectors, then, the suboptimal voltage vector vopt2 is selected to calculate the time of vector action, the other suboptimal voltage vector V_opt2 is selected to calculate the time of vector action. Finally, the corresponding modulation pulse is output to APF and this style reduces the switching frequency.

In this paper, another multi-vector MPCC strategy is used for three-phase two-level APF, that is, two adjacent vectors and one zero vector are selected in six sectors, six virtual vectors V_1d~V_6d with adjustable amplitude and direction are synthesized. The value function through the six vectors by calculated online, and the optimal voltage vector is obtained, the principle is shown in Figure 6. The implementation of this method mainly includes three steps: 1) obtain the vector action time, 2) synthesize the expected vector, 3) solve the value function to get the optimal vector.

5.png

Figure 5. New multi-vector composite graph

4.1 Calculation of vector action time

The operating time of the voltage vector is obtained by deadbeat error control of d-q axis current change rate. From Eq. (6), the slope of d-q axis under the action of zero vector is Eq. (19).

${{\mathsf{g}}_{\mathsf{0}}}\mathsf{=}\frac{\mathbf{\tilde{X}}(k)-\mathbf{\tilde{X}}(k-1)}{{{T}_{s}}}\left| _{\mathsf{u}=\mathsf{0}} \right.$           (19)

where,

$\mathbf{g}_{0}=\left[\begin{array}{l}g_{0 d} \\ g_{0 q}\end{array}\right], \mathbf{u}=\left[\begin{array}{l}u_{c d} \\ u_{c q}\end{array}\right]$             (20)

If V_n and V_m are two adjacent effective vectors, the d-q axis current slope of these two adjacent effective vectors is Eqns. (21) and (22), and [V_nd;V_nq] and [V_md;V_mq] are the d-q axis components of V_n and V_m.

${{\mathsf{g}}_{n}}=\frac{\mathbf{\tilde{X}}(k)-\mathbf{\tilde{X}}(k-1)}{{{T}_{s}}}\left| _{\mathsf{u}=\left[ \begin{smallmatrix}  {{\operatorname{V}}_{\operatorname{n}d}} \\  {{\operatorname{V}}_{\operatorname{n}q}} \end{smallmatrix} \right]} \right.$               (21)

${{\mathsf{g}}_{m}}=\frac{\mathbf{\tilde{X}}(k)-\mathbf{\tilde{X}}(k-1)}{{{T}_{s}}}\left| _{\mathsf{u}=\left[ \begin{smallmatrix}  {{\operatorname{V}}_{\operatorname{m}d}} \\  {{\operatorname{V}}_{\operatorname{m}q}} \end{smallmatrix} \right]} \right.$          (22)

where, [g_nd;g_nq] and [g_md;g_mq] are the slopes of the d-q axis components of vector V_n and V_m, respectively

$\mathbf{g}_{n}=\left[\begin{array}{l}g_{n d} \\ g_{n q}\end{array}\right], \mathbf{g}_{m}=\left[\begin{array}{l}g_{m d} \\ g_{m q}\end{array}\right]$          (23)

According to the principle of dead-beat control, there is no error between the predicted value and the given value at k + 1 time. Therefore, the discrete prediction model (24) can be obtained, t_n, t_m, t_o are the action time of the expected vector V_nd, V_md, and zero vector, the purpose is to predict that the output value is equal to the harmonic reference value, that is, zero error.

${{\mathsf{e}}_{ss}}=\mathbf{\tilde{X}}(k+1)-{{C}_{dq}}{{C}_{32}}{{\mathbf{\tilde{X}}}_{ref}}(k+1)=\mathsf{0}$         (24)

${{\mathsf{e}}_{ss}}=\left[ \begin{align}  & {{e}_{ssd}} \\  & {{e}_{ssq}} \\ \end{align} \right]$                (25)

If we combine the above formula, we can get that t_n, t_m and t_o are in Eq. (26).

$\begin{align}  & {{t}_{n}}=\frac{{{e}_{ssd}}({{g}_{mq}}-{{g}_{0q}})+{{e}_{ssq}}({{g}_{0d}}-{{g}_{md}})}{{{g}_{0d}}({{g}_{nq}}-{{g}_{mq}})+{{g}_{0q}}({{g}_{md}}-{{g}_{nd}})+{{g}_{nq}}({{g}_{nd}}-{{g}_{md}})} \\  & \quad \ +\frac{{{T}_{s}}({{g}_{0q}}{{g}_{md}}-{{g}_{mq}}{{g}_{0d}})}{{{g}_{0d}}({{g}_{nq}}-{{g}_{mq}})+{{g}_{0q}}({{g}_{md}}-{{g}_{nd}})+{{g}_{nq}}({{g}_{nd}}-{{g}_{md}})} \\ \end{align}$

$\begin{align}  & {{t}_{m}}=\frac{{{e}_{ssd}}({{g}_{0q}}-{{g}_{nq}})+{{e}_{ssq}}({{g}_{nd}}-{{g}_{0d}})}{{{g}_{0d}}({{g}_{nq}}-{{g}_{mq}})+{{g}_{0q}}({{g}_{md}}-{{g}_{nd}})+{{g}_{nq}}({{g}_{nd}}-{{g}_{md}})} \\  & \quad \ +\frac{{{T}_{s}}({{g}_{0d}}{{g}_{nq}}-{{g}_{0q}}{{g}_{nd}})}{{{g}_{0d}}({{g}_{nq}}-{{g}_{mq}})+{{g}_{0q}}({{g}_{md}}-{{g}_{nd}})+{{g}_{nq}}({{g}_{nd}}-{{g}_{md}})} \\ \end{align}$                     (26)

${{t}_{0}}={{T}_{s}}-{{t}_{n}}-{{t}_{m}}$

In all sectors, when the APF output tracks the reference harmonic current, if the vector action time is too long or the sampling period Ts is small enough, there may be two effective vectors whose action time is longer than the sampling period. here,

$t_{n}^{*}=\frac{{{t}_{n}}}{{{t}_{n}}+{{t}_{m}}}{{T}_{s}}$

$t_{n}^{*}=\frac{{{t}_{m}}}{{{t}_{n}}+{{t}_{m}}}{{T}_{s}}$                  (27)

$t_{n}^{*}=0$

4.2 Composite expected virtual vector

As shown in Figure 6, the expected virtual vector is composed of two adjacent effective vectors and a zero vector, so six expected virtual vectors are synthesized in the I ~ VI sector respectively, and then the optimal virtual vector can be obtained by traversing the objective function. Assuming that the required virtual vector is VD, then according to the action time of two adjacent effective vectors and one zero vector, they are combined respectively, as shown in Figure 6.

6.png

Figure 6. Virtual vector composition

7.png

Figure 7. Expectation vector action diagram

The process of V_d action is shown in Figure 7. The voltage of the resultant expected vector in d-q coordinate system is Eq. (28), and [u_cdd,u_cdq] is the component of APF expected output voltage in d-q axis.

$\left[ \begin{align}  & {{u}_{cdd}} \\  & {{u}_{cdq}} \\ \end{align} \right]=\left[ \begin{matrix}  \frac{{{t}_{n}}}{{{T}_{s}}} & {}  \\   {} & \frac{{{t}_{n}}}{{{T}_{s}}}  \\\end{matrix} \right]\left[ \begin{matrix}   {{u}_{nd}}  \\   {{u}_{nq}}  \\\end{matrix} \right]+\left[ \begin{matrix}   \frac{{{t}_{m}}}{{{T}_{s}}} & {}  \\   {} & \frac{{{t}_{m}}}{{{T}_{s}}}  \\\end{matrix} \right]\left[ \begin{matrix}   {{u}_{md}}  \\   {{u}_{mq}}  \\\end{matrix} \right]$          (28)

4.3 Selection of optimal virtual vector

At this time, the virtual vector is brought in to get the new input, the objective function is the same as Eq. (15), and the objective function is brought in to loop the six expected virtual vectors.

$\left[ \begin{align}  & {{u}_{cdd}} \\  & {{u}_{cdq}} \\ \end{align} \right]=\left[ \begin{matrix}   \frac{{{t}_{n}}}{{{T}_{s}}} & {}  \\   {} & \frac{{{t}_{n}}}{{{T}_{s}}}  \\\end{matrix} \right]\left[ \begin{matrix}   {{u}_{nd}}  \\   {{u}_{nq}}  \\\end{matrix} \right]+\left[ \begin{matrix}   \frac{{{t}_{m}}}{{{T}_{s}}} & {}  \\   {} & \frac{{{t}_{m}}}{{{T}_{s}}}  \\\end{matrix} \right]\left[ \begin{matrix}   {{u}_{md}}  \\   {{u}_{mq}}  \\\end{matrix} \right]$          (29)

The specific algorithm is shown in appendix 1.

In this paper, the three-phase two-level APF is taken as the control object, and the application of multi-vector MPCC in tracking and compensating harmonic current generated by nonlinear load is studied. By synthesizing six virtual expected vectors with adjustable size and direction from two effective vectors and one zero vector, this method has the accurate tracking effect, fast traversal calculation, and reduces the computational burden of the system. When the load changes, the tracking performance can be maintained all the time, and the tracking error is small throughout the whole process, which improves the control accuracy. A new method was compared with the traditional control method, it can maintain lower THD and improve the performance of APF. The looping optimization is an important part of FCS-MPC, this method combines two directions of simplified optimization process and adjacent candidate vectors, which is proved theoretically and can also be proved by constructing the Lyapunov function. It is feasible in practical application, which can reduce the control delay and APF performance.