﻿ 风能转换系统多目标滑模预测优化控制
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 哈尔滨工程大学学报  2018, Vol. 39 Issue (4): 760-765  DOI: 10.11990/jheu.201610076 0

### 引用本文

LENG Xin, SONG Wenlong, WANG Rui, et al. Multiobjective optimization of wind energy conversion systems using predictive and sliding mode controls[J]. Journal of Harbin Engineering University, 2018, 39(4), 760-765. DOI: 10.11990/jheu.201610076.

### 文章历史

1. 东北林业大学 机电工程学院, 黑龙江 哈尔滨 150040;
2. 哈尔滨移动公司 无线优化室, 黑龙江 哈尔滨 150001;
3. 哈尔滨工程大学 自动化学院, 黑龙江 哈尔滨 150001

Multiobjective optimization of wind energy conversion systems using predictive and sliding mode controls
LENG Xin1, SONG Wenlong1, WANG Rui2, YOU Jiang3, GU Weihong1, JIA Heming1
1. College of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin 150040, China;
2. Wireless optimization Room, Harbin Mobile Communication Company, Harbin 150001, China;
3. College of Automation, Harbin Engineering University, Harbin 150001, China
Abstract: To optimize energy conversion and to reduce the fatigue load of the doubly fed induction generators of wind energy conversion systems (WECS), a mathematical model was established and a sliding mode prediction control was proposed.This method adopts dual-frequency loop multiobjective control, in which the low-frequency component of wind speed based on ARMA model prediction is introduced into the low-frequency loop.PI controlled speed corresponds to the optimal tip speed to ensure that the operating point is around the Optimal Regimes Characteristic (ORC) curve; the turbulent component of wind speed is introduced into the high frequency loop.The dynamic optimization is realized by the combination of predictive and sliding mode controls.Simulation results reveal that the dual-frequency loop sliding mode prediction control effectively avoids the impact of uncertainties on the system, achieves an ORC tracking under partial load, reduces the change of control input, reduces mechanical fatigue, and ensures the optimization and stability of the system.
Key words: wind energy conversion system (WECS)    doubly fed induction generator(DFIG)    sliding mode prediction control (SMPC)    multiobjective    high-frequency loop    low-frequency loop    ARMA model    partial load

1 WECS的数学模型

 Download: 图 1 风能转换系统结构框图 Fig. 1 Structural diagram of WECS systems

 $v\left( t \right) = {v_s}\left( t \right) + {v_t}\left( t \right)$ (1)

 $\lambda = \frac{{R{\mathit{\Omega }_1}}}{v}$ (2)

 ${P_{{\rm{Wt}}}} = \frac{1}{2}\rho {\rm{ \mathit{ π} }}{R^2}{v^3}{C_{\rm{p}}}\left( \lambda \right)$ (3)

 $J\frac{{{\rm{d}}{\mathit{\Omega }_{\rm{h}}}}}{{{\rm{d}}t}} = {T_{{\rm{mec}}}} - {T_{\rm{G}}}$ (4)

 ${T_{\rm{G}}} = \frac{3}{2}p{L_{\rm{m}}}\left( {{i_{{\rm{Sq}}}}{i_{{\rm{Rd}}}} - {i_{{\rm{Rq}}}}{i_{{\rm{Sd}}}}} \right)$ (5)

 $\left\{ \begin{array}{l} \frac{{{\rm{d}}{i_{{\rm{Sd}}}}}}{{{\rm{d}}t}} = \frac{{{V_{{\rm{Sd}}}}}}{{{L_{\rm{S}}}}} - \frac{{{R_{\rm{S}}}}}{{{L_{\rm{S}}}}}{i_{{\rm{Sd}}}} - \frac{{{L_{\rm{m}}}}}{{{L_{\rm{S}}}}}\frac{{{\rm{d}}{i_{{\rm{Rd}}}}}}{{{\rm{d}}t}} + {\omega _{\rm{S}}}\left( {{i_{{\rm{Sq}}}} + \frac{{{L_{\rm{m}}}}}{{{L_{\rm{S}}}}}{i_{{\rm{Rq}}}}} \right)\\ \frac{{{\rm{d}}{i_{{\rm{Sq}}}}}}{{{\rm{d}}t}} = \frac{{{V_{{\rm{Sq}}}}}}{{{L_{\rm{S}}}}} - \frac{{{R_{\rm{S}}}}}{{{L_{\rm{S}}}}}{i_{{\rm{Sq}}}} - \frac{{{L_{\rm{m}}}}}{{{L_{\rm{S}}}}}\frac{{{\rm{d}}{i_{{\rm{Rq}}}}}}{{{\rm{d}}t}} - {\omega _{\rm{S}}}\left( {{i_{{\rm{Sd}}}} + \frac{{{L_{\rm{m}}}}}{{{L_{\rm{S}}}}}{i_{{\rm{Rd}}}}} \right)\\ \frac{{{\rm{d}}{i_{{\rm{Rd}}}}}}{{{\rm{d}}t}} = \frac{{{V_{{\rm{Rd}}}}}}{{{L_{\rm{R}}}}} - \frac{{{R_{\rm{R}}}}}{{{L_{\rm{R}}}}}{i_{{\rm{Rd}}}} - \frac{{{L_{\rm{m}}}}}{{{L_{\rm{R}}}}}\frac{{{\rm{d}}{i_{{\rm{Sd}}}}}}{{{\rm{d}}t}} + \left( {{\omega _{\rm{S}}} - \omega } \right)\left( {{i_{{\rm{Rq}}}} + \frac{{{L_{\rm{m}}}}}{{{L_{\rm{R}}}}}{i_{{\rm{Sq}}}}} \right)\\ \frac{{{\rm{d}}{i_{{\rm{Rq}}}}}}{{{\rm{d}}t}} = \frac{{{V_{{\rm{Rq}}}}}}{{{L_{\rm{R}}}}} - \frac{{{R_{\rm{R}}}}}{{{L_{\rm{R}}}}}{i_{{\rm{Rq}}}} - \frac{{{L_{\rm{m}}}}}{{{L_{\rm{R}}}}}\frac{{{\rm{d}}{i_{{\rm{Sq}}}}}}{{{\rm{d}}t}} - \left( {{\omega _{\rm{S}}} - \omega } \right)\left( {{i_{{\rm{Rd}}}} + \frac{{{L_{\rm{m}}}}}{{{L_{\rm{R}}}}}{i_{{\rm{Sd}}}}} \right) \end{array} \right.$ (6)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{x}} = {\left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)}&{{x_2}\left( t \right)}&{{x_3}\left( t \right)}&{{x_4}\left( t \right)} \end{array}} \right]^{\rm{T}}} = \\ \;\;\;\;\;\;{\left[ {\begin{array}{*{20}{c}} {{i_{{\rm{Sd}}}}}&{{i_{{\rm{Sq}}}}}&{{i_{{\rm{Rd}}}}}&{{i_{{\rm{Rq}}}}} \end{array}} \right]^{\rm{T}}}\\ \mathit{\boldsymbol{u}} = {\left[ {\begin{array}{*{20}{c}} {{V_{{\rm{Sd}}}}}&{{V_{{\rm{Sq}}}}}&{{V_{{\rm{Rd}}}}}&{{V_{{\rm{Rq}}}}} \end{array}} \right]^{\rm{T}}} \end{array} \right.$

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{A}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_{\rm{h}}}} \right)\mathit{\boldsymbol{x}} + \mathit{\boldsymbol{Bu}}\\ y = {T_{\rm{G}}} = \frac{{3p{L_{\rm{m}}}}}{2}\left( {{x_2}{x_3} - {x_1}{x_4}} \right) \end{array} \right.$ (7)

σ=1-Lm2/(LSLR), 则

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{A}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_{\rm{h}}}} \right) = \left[ {\begin{array}{*{20}{c}} { - \frac{{{R_{\rm{S}}}}}{{\sigma {L_{\rm{S}}}}}}&{{\omega _{\rm{S}}} + \frac{{p{\mathit{\Omega }_{\rm{h}}}L_{\rm{m}}^2}}{{\sigma {L_{\rm{S}}}{L_{\rm{R}}}}}}&{\frac{{{L_{\rm{m}}}{R_{\rm{R}}}}}{{\sigma {L_{\rm{S}}}{L_{\rm{R}}}}}}&{\frac{{p{\mathit{\Omega }_{\rm{h}}}{L_{\rm{m}}}}}{{\sigma {L_{\rm{S}}}}}}\\ { - \left[ {{\omega _{\rm{S}}} + \frac{{p{\mathit{\Omega }_{\rm{h}}}L_{\rm{m}}^2}}{{\sigma {L_{\rm{S}}}{L_{\rm{R}}}}}} \right]}&{ - \frac{{{R_{\rm{S}}}}}{{\sigma {L_{\rm{S}}}}}}&{ - \frac{{p{\mathit{\Omega }_{\rm{h}}}{L_{\rm{m}}}}}{{\sigma {L_{\rm{S}}}}}}&{\frac{{{L_{\rm{m}}}{R_{\rm{R}}}}}{{\sigma {L_{\rm{S}}}{L_{\rm{R}}}}}}\\ {\frac{{{L_{\rm{m}}}{R_{\rm{R}}}}}{{\sigma {L_{\rm{S}}}{L_{\rm{R}}}}}}&{ - \frac{{p{\mathit{\Omega }_{\rm{h}}}{L_{\rm{m}}}}}{{\sigma {L_{\rm{R}}}}}}&{ - \frac{{{R_{\rm{R}}}}}{{\sigma {L_{\rm{R}}}}}}&{{\omega _{\rm{S}}} - \frac{{p{\mathit{\Omega }_{\rm{h}}}}}{\sigma }}\\ {\frac{{p{\mathit{\Omega }_{\rm{h}}}{L_{\rm{m}}}}}{{\sigma {L_{\rm{R}}}}}}&{\frac{{{R_{\rm{S}}}{L_{\rm{m}}}}}{{\sigma {L_{\rm{S}}}{L_{\rm{R}}}}}}&{\frac{{p{\mathit{\Omega }_{\rm{h}}}}}{\sigma } - {\omega _{\rm{S}}}}&{ - \frac{{{R_{\rm{S}}}}}{{\sigma {L_{\rm{S}}}}}} \end{array}} \right]\\ \mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {\frac{1}{{\sigma {L_{\rm{S}}}}}}&0&{ - \frac{{{L_{\rm{m}}}}}{{\sigma {L_{\rm{S}}}{L_{\rm{R}}}}}}&0\\ 0&{\frac{1}{{\sigma {L_{\rm{S}}}}}}&0&{ - \frac{{{L_{\rm{m}}}}}{{\sigma {L_{\rm{S}}}{L_{\rm{R}}}}}}\\ { - \frac{{{L_{\rm{m}}}}}{{\sigma {L_{\rm{S}}}{L_{\rm{R}}}}}}&0&{\frac{1}{{\sigma {L_{\rm{R}}}}}}&0\\ 0&{ - \frac{{{L_{\rm{m}}}}}{{\sigma {L_{\rm{S}}}{L_{\rm{R}}}}}}&0&{\frac{1}{{\sigma {L_{\rm{R}}}}}} \end{array}} \right] \end{array} \right.$
2 基于频率分离原则的风能转换系统优化控制

WECS有两个时间尺度的动态性能, 分别对应于风速动态模型中的两种频谱变化:风速模型中低频部分的低频动态性能和风速模型湍流部分的高频动态性能。

 Download: 图 2 双环优化控制结构框图 Fig. 2 Block diagram of double loop optimal control
2.1 低频环PI稳态优化

 $\mathit{\bar \Omega }_{\rm{h}}^{{\rm{opt}}} = {v_s}i{\lambda _{{\rm{opt}}}}/R$ (8)

vsf为LPF的输出, vsp是通过滤波和预测得到的vs的估计值, 利用ARMA模型的递归思想, 可以由vsf对第kvsp值进行预测:

 $v_{{s_k}}^p = \sum\limits_{i = 1}^n {{a_i}v_{{s_{k - i}}}^f} + \sum\limits_{j = 1}^m {{b_j}\left( {v_{{s_{k - j}}}^p - v_{{s_{k - j}}}^f} \right)}$ (9)
2.2 高频环滑模预测优化控制

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{\dot x}}\left( \mathit{\boldsymbol{t}} \right) = \mathit{\boldsymbol{A}}x\left( t \right) + \mathit{\boldsymbol{B}}u\left( t \right) + \mathit{\boldsymbol{L}}e\left( t \right)\\ z = \mathit{\boldsymbol{C}}x\left( t \right) \end{array} \right.$ (10)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{x}}\left( {k + 1} \right) = \mathit{\boldsymbol{Ax}}\left( k \right) + \mathit{\boldsymbol{Bu}}\left( k \right)\\ \mathit{\boldsymbol{y}}\left( k \right) = \mathit{\boldsymbol{Cx}}\left( k \right) \end{array} \right.$ (11)

 $s\left( k \right) = Cx\left( k \right)$ (12)

 ${s_y}\left( {k + p} \right) = \mathit{\boldsymbol{C}}{\mathit{\boldsymbol{A}}^p}\mathit{\boldsymbol{x}}\left( k \right) + \sum\limits_{i = 1}^p {\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{A}}^{i - 1}}\mathit{\boldsymbol{Bu}}\left( {k + p - i} \right)}$ (13)

 $\begin{array}{*{20}{c}} {{{\hat s}_y}\left( {k + p} \right) = {s_y}\left( {k + p} \right) + {h_p}\left[ {s\left( k \right) - {s_y}\left( k \right)} \right] = }\\ {\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{A}}^p}\mathit{\boldsymbol{x}}\left( k \right) + \sum\limits_{i = 1}^p {\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{A}}^{i - 1}}\mathit{\boldsymbol{Bu}}\left( {k + p - i} \right)} + }\\ {{h_p}\left[ {s\left( k \right) - {s_y}\left( k \right)} \right]} \end{array}$ (14)

 ${s_y}\left( k \right) = \mathit{\boldsymbol{C}}{\mathit{\boldsymbol{A}}^p}\mathit{\boldsymbol{x}}\left( {k - p} \right) + \sum\limits_{i = 1}^p {\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{A}}^{i - 1}}\mathit{\boldsymbol{Bu}}\left( {k - i} \right)}$

 $\begin{array}{*{20}{c}} {{J_p} = \sum\limits_{i = 1}^N {{q_i}{{\left[ {{{\hat s}_y}\left( {k + i} \right) - {s_r}\left( {k + i} \right)} \right]}^2}} + }\\ {\sum\limits_{j = 1}^M {{r_j}{{\left[ {u\left( {k + j - 1} \right)} \right]}^2}} } \end{array}$ (15)

 $\left\{ \begin{array}{l} \begin{array}{*{20}{c}} {{s_r}\left( {k + p} \right) = \left( {1 - qT} \right){s_r}\left( {k + p - 1} \right) - }\\ {\varepsilon T{\mathop{\rm sgn}} \left( {{s_r}\left( {k + p} \right)} \right)} \end{array}\\ {s_r}\left( k \right) = s\left( k \right) \end{array} \right.$ (16)

 $\begin{array}{*{20}{c}} {{J_p} = {{\left[ {{{\mathit{\boldsymbol{\hat S}}}_y}\left( {k + 1} \right) - {\mathit{\boldsymbol{S}}_r}\left( {k + 1} \right)} \right]}^{\rm{T}}}\mathit{\boldsymbol{Q}}\left[ {{{\mathit{\boldsymbol{\hat S}}}_y}\left( {k + 1} \right) - } \right.}\\ {\left. {{\mathit{\boldsymbol{S}}_r}\left( {k + 1} \right)} \right] + {\mathit{\boldsymbol{U}}^{\rm{T}}}\left( k \right)\mathit{\boldsymbol{RU}}\left( k \right) = \left[ {\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}x\left( k \right) + \mathit{\boldsymbol{GU}}\left( k \right) + } \right.}\\ {{{\left. {\mathit{\boldsymbol{HE}}\left( k \right) - {\mathit{\boldsymbol{S}}_r}\left( {k + 1} \right)} \right]}^{\rm{T}}}\mathit{\boldsymbol{Q}}\left[ {\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}x\left( k \right) + \mathit{\boldsymbol{GU}}\left( k \right) + } \right.}\\ {\left. {\mathit{\boldsymbol{HE}}\left( k \right) - {\mathit{\boldsymbol{S}}_r}\left( {k + 1} \right)} \right] + {\mathit{\boldsymbol{U}}^{\rm{T}}}\left( k \right)\mathit{\boldsymbol{RU}}\left( k \right)} \end{array}$ (17)

 ${{\mathit{\boldsymbol{\hat S}}}_r}\left( {k + 1} \right) = {\left[ {{{\hat s}_y}\left( {k + 1} \right),{{\hat s}_y}\left( {k + 2} \right), \cdots ,{{\hat s}_y}\left( {k + N} \right)} \right]^{\rm{T}}},$
 ${\mathit{\boldsymbol{S}}_r}\left( {k + 1} \right) = {\left[ {{s_r}\left( {k + 1} \right),{s_r}\left( {k + 2} \right), \cdots ,{s_r}\left( {k + N} \right)} \right]^{\rm{T}}},$
 $\mathit{\boldsymbol{U}}\left( k \right) = {\left[ {u\left( k \right),u\left( {k + 1} \right), \cdots ,u\left( {k + M - 1} \right)} \right]^{\rm{T}}},$
 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{E}}\left( k \right) = \left[ {s\left( k \right) - {s_y}\left( {k\left| {k - 1} \right.} \right),s\left( k \right) - {s_y}\left( {k\left| {k - 2} \right.} \right),} \right.}\\ {{{\left. { \cdots ,s\left( k \right) - {s_y}\left( {k\left| {k - N} \right.} \right)} \right]}^{\rm{T}}},} \end{array}$
 $\mathit{\boldsymbol{Q}} = {\rm{diag}}\left( {{q_1},{q_2}, \cdots ,{q_N}} \right),$
 $\mathit{\boldsymbol{R}} = {\rm{diag}}\left( {{r_1},{r_2}, \cdots ,{r_M}} \right),$
 $\mathit{\boldsymbol{H}} = {\rm{diag}}\left( {{h_1},{h_2}, \cdots ,{h_N}} \right),$
 $\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{CA}}}\\ {\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{A}}^2}}\\ \mathit{\boldsymbol{M}}\\ {\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{A}}^N}} \end{array}} \right];\mathit{\boldsymbol{G}} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{CB}}}&\mathit{\boldsymbol{0}}&\mathit{\boldsymbol{L}}&\mathit{\boldsymbol{0}}\\ {\mathit{\boldsymbol{CAB}}}&{\mathit{\boldsymbol{CB}}}&\mathit{\boldsymbol{L}}&\mathit{\boldsymbol{0}}\\ \mathit{\boldsymbol{M}}&\mathit{\boldsymbol{M}}&\mathit{\boldsymbol{M}}&\mathit{\boldsymbol{M}}\\ {\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{A}}^{N - 1}}\mathit{\boldsymbol{B}}}&{\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{A}}^{N - 2}}\mathit{\boldsymbol{B}}}&\mathit{\boldsymbol{L}}&{\mathit{\boldsymbol{CB}}} \end{array}} \right]$

∂Jp/∂U(k)=0, 可得:

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{U}}\left( k \right) = {{\left( {{\mathit{\boldsymbol{G}}^{\rm{T}}}\mathit{\boldsymbol{QG}} + \mathit{\boldsymbol{R}}} \right)}^{ - 1}}{\mathit{\boldsymbol{G}}^{\rm{T}}}\mathit{\boldsymbol{Q}}\left[ {{\mathit{\boldsymbol{S}}_r}\left( {k + 1} \right) - } \right.}\\ {\left. {\mathit{\boldsymbol{ \boldsymbol{\varPhi} x}}\left( k \right) - \mathit{\boldsymbol{HE}}\left( k \right)} \right]} \end{array}$ (18)

 $\begin{array}{*{20}{c}} {u\left( k \right) = \left[ {1,0, \cdots ,0} \right]{{\left( {{\mathit{\boldsymbol{G}}^{\rm{T}}}\mathit{\boldsymbol{QG}} + \mathit{\boldsymbol{R}}} \right)}^{ - 1}}{\mathit{\boldsymbol{G}}^{\rm{T}}}\mathit{\boldsymbol{Q}}\left[ {{\mathit{\boldsymbol{S}}_r}\left( {k + } \right.} \right.}\\ {\left. {\left. 1 \right) - \mathit{\boldsymbol{ \boldsymbol{\varPhi} }}x\left( k \right) - \mathit{\boldsymbol{HE}}\left( k \right)} \right]} \end{array}$ (19)
2.3 基于WECS的频率分离原则的双环滑模预测控制算法

1) 由风速计测量出风速v的大小, 由编码器测量得到高速轴的转速Ωh;

2) 通过对风速v进行预测得到低频风速v;

3) 求得风速湍流分量Δv=v-v和其标准化值Δvv/v;

4) 对信号Ωh进行低通滤波得到高速轴转速的低频分量Ωh;

5) 求得高速轴转速的高频分量的标准化值ΔΩh=(Ωh-Ωh)/Ωh;

6) vΩh作为低频环的输入信号, 控制输入的稳态分量TG是稳态优化的输出;

7)Δv和ΔΩh作为高频环的输入, 高频环的设计基于上述滑模预测控制算法, 可以得到控制输入的动态分量ΔTG;

8) 总控制输入TG*通过稳态分量和动态分量相加得到:TG*=TGTG

3 仿真结果

 Download: 图 3 风速预测和低频滤波效果对比 Fig. 3 Comparison of wind speed prediction and low frequency filtering effect
 Download: 图 4 风力机的叶尖速度比 Fig. 4 Tip speed ratio of wind turbine
 Download: 图 5 叶尖速度比的标准变化 Fig. 5 The normalized change of tip speed ratia
 Download: 图 6 电磁转矩的标准变化 Fig. 6 The normalized change of electromagnetic torque
 Download: 图 7 功率系数的变化过程 Fig. 7 The change process of power coefficient
 Download: 图 8 工作点与ORC的相对关系 Fig. 8 The relative relationship between the operating point and the ORC
4 结论

1) 本文通过分析低功率刚性传动链双馈感应发电机的风能转换系统的特点, 建立了WECS的数学模型, 基于WECS的两个时间尺度的动态性能, 利用双频环优化控制结构解决了系统效率的最大化与控制输入投入的最小化的平衡问题。

2) 利用ARMA模型对低通滤波器分离出的风速进行预测, 实现了vs(t)的准确估计, 该方法能够减少低频分量的延迟。

3) 为弥补双频环控制结构对参数、模型不确定的敏感性, 提出高频环滑模预测控制方法。该方法充分利用滑模的鲁棒性和预测控制的滚动优化与反馈校正思想, 有效的避免了滑模控制高频切换带来的抖振现象, 在部分负荷状态下减小了控制输入量的变化, 降低了机械疲劳, 提高了能源利用效率。

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