Prediction error compensation method of FCSMPC for converter based on neural network

Abstract

FCSMPC is a classical converter predictive control algorithm whose control performance is affected by the prediction error of the prediction model. In classical predictive control theory, the feedback correction mechanism is used to compensate for such prediction error. However, when this strategy is directly applied to the FCSMPC algorithm, the prediction error cannot be easily calculated. To address the prediction error compensation problem of FCSMPC, this paper proposes a prediction error compensation method based on neural network. A neural network prediction model is also constructed based on the timing characteristics of prediction error signals. The prediction error of this prediction model at the next moment is calculated by the designed neural network model, and then the output of the prediction model is compensated at the current moment. To improve the anti-interference performance of FCSMPC, the MRSVD algorithm is used to filter the prediction error sample data and the neural networks are trained by these sample data. The adaptability of the prediction error calculation is further improved by combining offline training with the online calculation of the neural network. A simulation model of the proposed method is then constructed using MATLAB, and simulation results show that the control performance of the FCSMPC algorithm is improved and that the effectiveness and feasibility of the proposed method are verified.

Appendices

Appendix A

Figure 1 shows a three-phase inverter with an output LC filter. The converter and filter models are presented in this section, and the load is assumed to be unknown.

The switching states of the converter are determined by the gating signals S_a, S_b, and S_c as follows:

$$ S_{a} = \left\{ {\begin{array}{*{20}c} {1,\,{\text{if}}\,{\text{S}}_{{{\text{ap}}}} \,{\text{on}}\,{\text{and}}\,{\text{S}}_{{{\text{an}}}} \,{\text{off}}} \\ {0,\,{\text{if}}\,{\text{S}}_{{{\text{ap}}}} \,{\text{off}}\,{\text{and}}\,{\text{S}}_{{{\text{an}}}} \,{\text{on}}} \\ \end{array} } \right. $$

(9)

$$ S_{b} = \left\{ {\begin{array}{*{20}c} {1,\,{\text{if}}\,{\text{S}}_{{{\text{bp}}}} \,{\text{on}}\,{\text{and}}\,{\text{S}}_{{{\text{bn}}}} \,{\text{off}}} \\ {0,\,{\text{if}}\,{\text{S}}_{{{\text{bp}}}} \,{\text{off}}\,{\text{and}}\,{\text{S}}_{{{\text{bn}}}} \,{\text{on}}} \\ \end{array} } \right. $$

(10)

$$ S_{c} = \left\{ {\begin{array}{*{20}c} {1,\,{\text{if}}\,{\text{S}}_{{{\text{cp}}}} \,{\text{on}}\,{\text{and}}\,{\text{S}}_{{{\text{cn}}}} \,{\text{off}}} \\ {0,\,{\text{if}}\,{\text{S}}_{{{\text{cp}}}} \,{\text{off}}\,{\text{and}}\,{\text{S}}_{{{\text{cn}}}} \,{\text{on}}} \\ \end{array} } \right. $$

(11)

The switching function combination S can be expressed in vectorial form as

$$ S = \frac{2}{3}S_{{\text{a}}} + aS_{{\text{b}}} + a^{2} S_{{\text{c}}} $$

(12)

where the operator is α = exp(2πj/3).

The output–voltage space vectors generated by the inverter are defined as

$$ v_{i} = \frac{2}{3}\left( {v_{{\text{a}}} + \alpha v_{{\text{b}}} + \alpha^{2} v_{{\text{c}}} } \right) $$

(13)

where v_a, v_b, and v_c are the phase voltages of the inverter with respect to the negative terminal of the DC-link. The voltage vector v_i can be related to the switching state vector S by

$$ v_{{\text{i}}} = V_{{{\text{dc}}}} S $$

(14)

where V_dc is the DC-link voltage.

By considering all possible combinations of the gating signals S_a, S_b, and S_c, eight switching states and eight voltage vectors v_i(i = 0, 1, …, 7) are obtained. Note that v₀ = v₇, thereby resulting in only seven voltage vectors.

The inverter can be modeled as a continuous system by using modulation techniques, such as pulse-width modulation. Nevertheless, in this article, the inverter is considered a non-linear discrete system with only seven voltage vectors as possible outputs.

By using vectorial notation, the filter current i_s, output voltage v_c, and output current i_o can be expressed as space vectors and defined as

$$ i_{{\text{s}}} = \frac{2}{3}\left( {i_{{{\text{sa}}}} + \alpha i_{{{\text{sb}}}} + \alpha^{2} i_{{{\text{sc}}}} } \right) $$

(15)

$$ v_{{\text{c}}} = \frac{2}{3}\left( {v_{{{\text{ca}}}} + \alpha v_{{{\text{cb}}}} + \alpha^{2} v_{{{\text{cc}}}} } \right) $$

(16)

$$ i_{{\text{o}}} = \frac{2}{3}\left( {i_{{{\text{oa}}}} + \alpha i_{{{\text{ob}}}} + \alpha^{2} i_{{{\text{oc}}}} } \right) $$

(17)

The LC filter can be described by using two equations, with the first equation describing the inductance dynamics and the second equation describing the capacitor dynamics.

The filter inductance expressed in vectorial form is

$$ L\frac{{di_{{\text{s}}} }}{dt} = v_{{\text{i}}} - v_{{\text{c}}} $$

(18)

where L is the filter inductance.

The dynamic behavior of the output voltage can be expressed as

$$ C\frac{{dv_{{\text{c}}} }}{dt} = i_{{\text{s}}} - i_{{\text{o}}} $$

(19)

where C is the filter capacitance.

The above equations can be rewritten as the following state-space system:

$$ \frac{dx}{{dt}} = Ax + Bv_{{\text{i}}} + B_{{\text{d}}} i_{{\text{o}}} $$

(20)

where $x(k) = \left[ \begin{gathered} i_{s} (k) \hfill \\ v_{o} (k) \hfill \\ \end{gathered} \right]$ represents the state quantities, $A = \left[ {\begin{array}{*{20}c} { - r/L} & { - 1/L} \\ {1/C} & 0 \\ \end{array} } \right]$ is the system matrix, B = [1/L 0]^T is the input matrix, and B_d = [0 − 1/C]^T is the disturbance input matrix.

Variables i_s and v_c are measured, v_i can be calculated using Eq. (14), and i_o is considered an unknown disturbance. The value of V_dc is assumed to be fixed and known.

The output of the system is the output voltage v_c, which is written as the following state equation:

$$ v_{c} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ \end{array} } \right]x $$

(21)

A discrete-time model of the filter is obtained from Eq. (20) for a sampling time T_s and is expressed as

$$ x\left( {k + 1} \right) = A_{{\text{q}}} x\left( k \right) + B_{{\text{q}}} v_{{\text{s}}} \left( k \right) + B_{{{\text{dq}}}} i_{{\text{o}}} \left( k \right) $$

(22)

where $A_{{\text{q}}} = e^{{{\text{AT}}_{{\text{s}}} }} ,B_{q} = \int\limits_{0}^{{T_{{\text{s}}} }} {e^{A\tau } Bd\tau }$, $B_{{{\text{dq}}}} = \int\limits_{0}^{{{\text{T}}_{{\text{s}}} }} {e^{A\tau } B_{{\text{d}}} d\tau }$,

The above equations are used as predictive models in the proposed predictive controller.

Appendix B

Parameters of inverter: v_dc = 520 V; L_f = 2.4 mH; r_f = 0.05 Ω; C_f = 40 μF; and T_s = 40 us.

Parameters of load: three-phase resistive-inductive load with active power P = 15 kW and reactive power Q = 2 kvar.

Parameters of inverter: T_s = 40 us; λ_c = 0.7; and v_o^* is the three-phase AC voltage with a frequency of 50 Hz and amplitude of 200 V.