CN103248052B

CN103248052B - Saturated switching control method for three-phase parallel active power filter

Info

Publication number: CN103248052B
Application number: CN201310182664.XA
Authority: CN
Inventors: 程新功; 宗西举; 裴兴华; 张静亮; 侯广松; 白万建; 徐珂; 殷文月; 李石清; 王成友; 任宏伟; 王玉真; 邵振振; 于明珠; 丁冬睿; 陈早军; 张庆华; 张步胜; 欧朱建; 张梦华
Original assignee: State Grid Corp of China SGCC; University of Jinan; Heze Power Supply Co of State Grid Shandong Electric Power Co Ltd
Current assignee: State Grid Corp of China SGCC; University of Jinan; Heze Power Supply Co of State Grid Shandong Electric Power Co Ltd
Priority date: 2013-05-16
Filing date: 2013-05-16
Publication date: 2015-04-08
Anticipated expiration: 2033-05-16
Also published as: CN103248052A

Abstract

The invention relates to a saturation switching control method of a three-phase parallel active power filter, comprising the steps of: 1) establishing a switching system model according to each working mode of the three-phase parallel active power filter; 2) constructing a common Lyapunov function to obtain switching rules based on state feedback Among them, arg means to take the subscript that satisfies the expression; 3) When the three-phase parallel active power filter has saturated nonlinear constraints, an additional matrix G is introduced to linearize the saturated nonlinear constraints with the convex domain method, and then Lyapunuo The saturation switching rule is designed to make the saturated system asymptotically stable based on the stability theory of the husband function; 4) Under the action of the saturated switching rule, the three-phase parallel active power filter realizes the compensation of the harmonic current of the grid.

Description

A Saturation Switching Control Method of Three-phase Parallel Active Power Filter

技术领域technical field

本发明涉及一种三相并联型有源电力滤波器的饱和切换控制方法。The invention relates to a saturation switching control method of a three-phase parallel active power filter.

背景技术Background technique

随着电网中非线性负荷的日益增加，其产生的谐波对配电网造成了严重的负面影响。其间造成各种电气设备不能正常工作、附加谐波损耗、电压畸变、通信干扰等问题。有源电力滤波器被认为是改善电能质量、抑制谐波最有效的设备。With the increasing of non-linear loads in the power grid, the harmonics generated by it have a serious negative impact on the distribution network. In the meantime, various electrical equipment cannot work normally, additional harmonic loss, voltage distortion, communication interference and other problems. Active power filter is considered to be the most effective device for improving power quality and suppressing harmonics.

传统的有源电力滤波器都是以周期平均模型来分析和设计，但是这种近似得到小信号模型的建模方法在存在大信号干扰时，系统会出现不稳定或者补偿效果较差等问题。同时，在实际的APF控制系统中普遍存在饱和非线性限制环节。例如，系统发生短路故障或非线性负载突增时，为保证APF安全运行，小容量的APF(有源滤波器)通常采用限流保护策略(截断限流或按容量极限比例限流策略)。然而，常用的截断限流保护方式中通过饱和非线性限制环节来限制指令谐波电流，该环节可能会使系统的稳定性及动态性能发生变化甚至会导致严重后果。The traditional active power filter is analyzed and designed based on the cycle average model, but this modeling method of approximating the small signal model will cause problems such as instability or poor compensation effect in the presence of large signal interference. At the same time, in the actual APF control system, there is a saturated nonlinear limiting link. For example, when a short-circuit fault or a nonlinear load surge occurs in the system, in order to ensure the safe operation of the APF, the small-capacity APF (active filter) usually adopts a current-limiting protection strategy (cut-off current-limiting or current-limiting strategy proportional to capacity limit). However, in the commonly used truncation current limiting protection method, the command harmonic current is limited through the saturated nonlinear limiting link, which may change the stability and dynamic performance of the system and even lead to serious consequences.

发明内容Contents of the invention

本发明的目的就是为了解决现有有源电力滤波器传统控制方法的精度不高和饱和非线性的稳定对系统稳定的影响等问题。为此本发明提供了一种三相并联型有源电力滤波器的饱和切换控制方法，它具有良好的动态特性和稳态特性。The purpose of the invention is to solve the problems of low precision of the traditional control method of the existing active power filter and the influence of saturation nonlinear stability on system stability and the like. Therefore, the present invention provides a saturation switching control method of a three-phase parallel active power filter, which has good dynamic characteristics and steady-state characteristics.

为实现上述目的，本发明采用了如下技术方案：To achieve the above object, the present invention adopts the following technical solutions:

一种三相并联型有源电力滤波器的饱和切换控制方法，包括以下步骤：A method for controlling saturation switching of a three-phase parallel active power filter, comprising the following steps:

步骤1)首先分析三相并联型有源电力滤波器各工作模态拓扑结构，根据实际电路参数，建立三相有源电力滤波器的切换系统模型其中A_σ(t)为有源电力滤波器工作模态的系数矩阵，B为输入矩阵且是常数阵，这两个矩阵都是由实际电路参数决定，x为状态向量，由三相并联型有源电力滤波器的输出补偿电流和直流侧电压组成，u是输入向量，由系统电源电压组成；Step 1) First analyze the topological structure of each working mode of the three-phase parallel active power filter, and establish the switching system model of the three-phase active power filter according to the actual circuit parameters Among them, A _σ(t) is the coefficient matrix of the working mode of the active power filter, B is the input matrix and is a constant matrix, these two matrices are determined by the actual circuit parameters, x is the state vector, and the three-phase parallel type The output compensation current of the active power filter is composed of the DC side voltage, and u is the input vector, which is composed of the system power supply voltage;

步骤2)针对三相有源电力滤波器的切换系统模型设计模型的切换规则σ(t)，使得系统在切换平衡点渐进稳定，即实现了三相有源电力滤波器的电压稳定和电流跟踪的统一控制；首先根据线性系统理论的方法计算系统的公共李雅普诺夫函数V(ξ)＝ξ^TPξ,(ξ＝x-x_e)，x_e为系统的切换平衡点，即求解其中Q是单位矩阵，A_λ为Hurwitz矩阵，λ_i∈(0,1)，根据该李雅普诺夫函数得到切换规则 $σ (t) = \arg \min_{i &Element; N} ξ^{T} P (A_{i} x + Bu),$ 式中，arg表示下标；Step 2) Switched system model for three-phase active power filter The switching rule σ(t) of the design model makes the system asymptotically stable at the switching balance point, that is, the unified control of voltage stability and current tracking of the three-phase active power filter is realized; firstly, the common Lyapunov function V(ξ)=ξ ^T Pξ,(ξ=xx _e ), x _e is the switching equilibrium point of the system, that is, to solve where Q is the identity matrix, A _λ is the Hurwitz matrix, λ _i ∈ (0,1), According to the Lyapunov function, the switching rule is obtained $σ (t) = \arg \min_{i &Element; N} ξ^{T} P (A_{i} x + Bu),$ In the formula, arg represents the subscript;

步骤3)根据上述方法得到的控制规则，控制三相有源电力滤波器件的开关通断，实现对电网谐波电流的补偿。Step 3) According to the control rule obtained by the above method, control the on-off of the switch of the three-phase active power filter device, and realize the compensation for the harmonic current of the power grid.

所述步骤1)的具体步骤为：The concrete steps of described step 1) are:

(1)根据三相APF的工作原理即通过控制6个开关器件S₁～S₆的通断，使补偿电流i_cj跟踪指令电流保证了电网侧只含有正弦有功分量；同时使直流侧电容电压v_dc稳定于参考电压 (1) According to the working principle of the three-phase APF, the compensation current i _cj can track the command current by controlling the on-off of the six switching devices S ₁ ~ S ₆ It ensures that the grid side only contains sinusoidal active components; at the same time, the capacitor voltage v _dc on the DC side is stabilized at the reference voltage

基于基尔霍夫定律，建立三相有源电力滤波器的电压电流方程：Based on Kirchhoff's law, the voltage and current equation of the three-phase active power filter is established:

$\{\begin{matrix} {e e}_{a a} = = L L \frac{{di di}_{ca ca}}{dt dt} + + {Ri Ri}_{ca ca} + + {U u}_{an an} + + {U u}_{nN n} \\ {e e}_{b b} = = L L \frac{{di di}_{cb cb}}{dt dt} + + {Ri Ri}_{cb cb} + + {U u}_{bn bn} + + {U u}_{nN n} \\ {e e}_{c c} = = L L \frac{{di di}_{cc cc}}{dt dt} + + {Ri Ri}_{cc cc} + + {U u}_{cn cn} + + {U u}_{nN n} \end{matrix} - - - - - - ((11))$

$\frac{{di di}_{cj cj}}{dt dt} = = - - \frac{R R}{L L} {i i}_{cj cj} - - \frac{11}{L L} (({p p}_{j j} - - \frac{11}{33} \underset{j j = = a a,, b b,, c c}{Σ Σ} {p p}_{j j})) {v v}_{dc dc} + + \frac{{e e}_{j j}}{L L},, j j = = a a,, b b,, c c - - - - - - ((22))$

$C C \frac{{dv dv}_{dc dc}}{dt dt} = = \underset{j j = = a a,, b b,, c c}{Σ Σ} {p p}_{j j} {i i}_{cj cj} - - - - - - ((33))$

p_j(j＝a,b,c)表示j相开关状态，L为三相滤波电感；C为直流侧电容；R为电感等值电阻；e_j(j＝a,b,c)为系统电源电压；v_dc为直流侧电容电压；i_Lj是非线性负荷电流；i_sj为电源电流；i_cj为三相APF的补偿电流；令表示负载电流中的谐波分量和无功分量；U_jn(j＝a,b,c)为j相和n点间的电压；U_nN是n和中性点N间的电压。p _j (j=a,b,c) represents the switch state of phase j, L is the three-phase filter inductance; C is the DC side capacitance; R is the equivalent resistance of the inductor; e _j (j=a,b,c) is the system power supply voltage; v _dc is the DC side capacitor voltage; i _Lj is the nonlinear load current; i _sj is the power supply current; i _cj is the compensation current of the three-phase APF; Indicates the harmonic component and reactive component in the load current; U _jn (j=a,b,c) is the voltage between phase j and point n; U _nN is the voltage between n and neutral point N.

定义S_j(j＝a,b,c)为开关函数，其定义如下：Define S _j (j=a,b,c) as a switch function, which is defined as follows:

${S S}_{j j} = = {p p}_{j j} - - \frac{11}{33} \underset{j j = = a a,, b b,, c c}{Σ Σ} {p p}_{j j},, j j = = a a,, b b,, c c - - - - - - ((44))$

则状态方程(1)和(3)可写为：Then the state equations (1) and (3) can be written as:

$[\begin{matrix} \frac{{di di}_{ca ca}}{dt dt} \\ \frac{{di di}_{cb cb}}{dt dt} \\ \frac{{di di}_{cc cc}}{dt dt} \\ \frac{{dv dv}_{dc dc}}{dt dt} \end{matrix}] = = [\begin{matrix} - - \frac{R R}{L L} & 00 & 00 & - - \frac{{S S}_{a a}}{L L} \\ 00 & - - \frac{R R}{L L} & 00 & - - \frac{{S S}_{b b}}{L L} \\ 00 & 00 & - - \frac{R R}{L L} & - - \frac{{S S}_{c c}}{L L} \\ \frac{{p p}_{a a}}{C C} & \frac{{p p}_{b b}}{C C} & \frac{{p p}_{c c}}{C C} & 00 \end{matrix}] [\begin{matrix} {i i}_{ca ca} \\ {i i}_{cb cb} \\ {i i}_{cc cc} \\ {v v}_{dc dc} \end{matrix}] + + [\begin{matrix} \frac{{e e}_{a a}}{L L} \\ \frac{{e e}_{b b}}{L L} \\ \frac{{e e}_{c c}}{L L} \\ 00 \end{matrix}] - - - - - - ((55))$

对三相有源电力滤波器而言，不存在某一桥的上下桥臂均关断的情况，用p_a，p_b，p_c对应6种开关组合，即：001、010、011、100、101、110，定义上述组合分别对应6种切换模态σ(t)∈N＝{1,2,…,6}。根据各工作模态的拓扑结构，式(5)的状态方程可写为：For a three-phase active power filter, there is no case where both the upper and lower bridge arms of a certain bridge are turned off, and pa, _p _b , and p _c correspond to six switch combinations, namely: 001, 010, 011, 100 , 101, 110, define that the above combinations correspond to six switching modes σ(t)∈N={1,2,...,6} respectively. According to the topological structure of each working mode, the state equation of formula (5) can be written as:

$\overset{\cdot \cdot}{x x} = = {A A}_{σ σ ((t t))} x x + + Bu Bu - - - - - - ((66))$

式中，APF的状态向量x＝[i_ca,i_cb,i_cc,v_dc]^T和输入向量u＝[e_a,e_b,e_c,0]^T；A，B为APF工作模态的系数矩阵；其中A_σ(t)∈{A₁,A₂,A₃,A₄,A₅,A₆}，对于输入矩阵B，其不受切换过程的影响，故保持不变；针对每一种开关组合，得到如的状态方程，其中：In the formula, APF state vector x=[i _ca ,i _cb ,i _cc ,v _dc ] ^T and input vector u=[e _a ,e _b ,e _c ,0] ^T ; A and B are APF working modes coefficient matrix; where A _σ(t) ∈{A ₁ ,A ₂ ,A ₃ ,A ₄ ,A ₅ ,A ₆ }, for the input matrix B, it is not affected by the switching process, so it remains unchanged; for For each switch combination, we get as The state equation of , where:

${A A}_{σ σ ((t t))} = = [\begin{matrix} - - \frac{R R}{L L} & 00 & 00 & - - \frac{11}{L L} (({p p}_{a a} - - \frac{11}{33} \underset{j j = = a a,, b b,, c c}{Σ Σ} {p p}_{j j})) \\ 00 & - - \frac{R R}{L L} & 00 & - - \frac{11}{L L} (({p p}_{b b} - - \frac{11}{33} \underset{j j = = a a,, b b,, c c}{Σ Σ} {p p}_{j j})) \\ 00 & 00 & - - \frac{R R}{L L} & - - \frac{11}{L L} (({p p}_{c c} - - \frac{11}{33} \underset{j j = = a a,, b b,, c c}{Σ Σ} {p p}_{j j})) \\ \frac{{p p}_{a a}}{C C} & \frac{{p p}_{b b}}{C C} & \frac{{p p}_{c c}}{C C} & 00 \end{matrix}],, B B = = [\begin{matrix} - - \frac{11}{L L} & 00 & 00 & 00 \\ 00 & - - \frac{11}{L L} & 00 & 00 \\ 00 & 00 & - - \frac{11}{L L} & 00 \\ 00 & 00 & 00 & 00 \end{matrix}] - - - - - - ((77)) . .$

所述步骤2)的具体步骤为：The concrete steps of described step 2) are:

切换平衡点x_e，为三相指令谐波电流及直流侧参考电压，记为指令谐波电流直接测到，直流侧参考电压为一给定的常量；根据三相有源电力滤波器6种工作模态，有A_σ(t)∈{A₁,A₂,A₃,A₄,A₅,A₆}；定义子系统的凸组合为其中λ_i∈Λ，如果凸组合A_λ为Hurwitz矩阵，即A_λ是稳定的，则存在P和Q使得， $A_{λ}^{T} P + {PA}_{λ} = - Q$ 成立。The switching balance point x _e is the three-phase command harmonic current and the reference voltage of the DC side, denoted as The command harmonic current is directly measured, and the reference voltage of the DC side is a given constant; according to the six working modes of the three-phase active power filter, there is A _σ(t) ∈{A ₁ ,A ₂ ,A ₃ , A ₄ ,A ₅ ,A ₆ }; define the convex combination of the subsystem as where λ _i ∈ Λ, If the convex combination A _λ is a Hurwitz matrix, that is, A _λ is stable, then there exist P and Q such that, $A_{λ}^{T} P + {PA}_{λ} = - Q$ established.

当步骤(2)所述的规则在实际工程应用中，为了提高有源电力滤波器对输出补偿电流的有效控制能力和过载能力，通常采用截断限流保护策略，使其安全运行，但是，在该策略中存在饱和非线性限制环节，该环节会使原本稳定系统失稳，因此根据李雅普诺夫稳定性理论以及凸域法，仅当所得正定对称阵P满足不等式When the rules described in step (2) are applied in practical engineering, in order to improve the effective control capability and overload capability of the output compensation current of the active power filter, the truncation current limiting protection strategy is usually adopted to make it operate safely. However, in There is a saturated nonlinear limitation link in this strategy, which will destabilize the originally stable system. Therefore, according to the Lyapunov stability theory and the convex domain method, only when The resulting positive definite symmetric matrix P satisfies the inequality

${(({D D.}_{i i} {A A}_{λ λ} + + {D D.}_{i i}^{- -} G G))}^{T T} P P + + P P (({D D.}_{i i} {A A}_{λ λ} + + {D D.}_{i i}^{- -} G G)) < < 00,, {D D.}_{i i} &Element; &Element; {D D.}_{n no},, i i = = 11,, 22,, . . . . . .,, 22^{n no} - - - - - - ((1414))$

时，有源电力滤波器可以在截断限流策略中稳定工作。When , the active power filter can work stably in the truncation current limiting strategy.

本发明的有益效果在于：The beneficial effects of the present invention are:

1)由于采用状态反馈的切换控制，实现了直流侧电压和电流补偿的统一控制，省去了单独的电压控制环节，从而使系统简单，避免了控制系统参数的整定。1) Due to the use of state feedback switching control, the unified control of DC side voltage and current compensation is realized, and a separate voltage control link is omitted, thereby making the system simple and avoiding the tuning of control system parameters.

2)饱和非线性限制的加入，解决了有源电力滤波器不具备对输出电流的过载抑制能力，同时引入的状态饱和稳定性设计，保证了系统的稳定性。2) The addition of the saturation nonlinear limit solves the problem that the active power filter does not have the ability to suppress the overload of the output current. At the same time, the introduction of the state saturation stability design ensures the stability of the system.

3)本发明不涉及复杂运算，控制算法简单，运行周期短。3) The present invention does not involve complicated calculation, the control algorithm is simple, and the operation period is short.

附图说明Description of drawings

图1为三相并联型有源电力滤波器的拓扑结构图；Figure 1 is a topological structure diagram of a three-phase parallel active power filter;

图2为有源电力滤波器的控制系统框图；Fig. 2 is a control system block diagram of an active power filter;

图3i_L1，i_L2及i_L波形；Figure 3i _L1 , i _L2 and i _L waveforms;

图4方式1下谐波参考电流及输出电流Figure 4 Mode 1 Harmonic reference current and output current

图5方式1下电网电流、直流侧电压波形；Figure 5 grid current and DC side voltage waveforms under Mode 1;

图6方式2下和i_c波形；Figure 6 Mode 2 and _ic waveform;

图7方式2下THD_i_L，THD_i_s，电网电流和直流侧电压。Figure 7 THD_i _L , THD_i _s , grid current and DC side voltage under mode 2.

具体实施方式Detailed ways

下面结合附图与实例对本发明作进一步说明。The present invention will be further described below in conjunction with accompanying drawings and examples.

三相APF的原理如图1所示。图中：L为三相滤波电感；C为直流侧电容；R为电感等值电阻；e_j(j＝a,b,c)为系统电源电压；v_dc为直流电容电压；i_Lj是非线性负荷电流；i_sj为电源电流；i_cj为三相APF的补偿电流；令表示负载电流中的谐波分量和无功分量。三相APF的工作原理是通过控制6个开关器件S₁～S₆的通断，使补偿电流icj跟踪指令电流保证了电网侧只含有正弦有功分量；同时使直流侧电压v_dc稳定于参考电压根据基尔霍夫定律，建立三相有源电力滤波器的电压电流方程：The principle of the three-phase APF is shown in Figure 1. In the figure: L is the three-phase filter inductor; C is the DC side capacitor; R is the equivalent resistance of the inductor; e _j (j=a,b,c) is the system power supply voltage; v _dc is the DC capacitor voltage; i _Lj is the nonlinear load current; i _sj is the power supply current; i _cj is the compensation current of the three-phase APF; let Indicates the harmonic component and reactive component in the load current. The working principle of the three-phase APF is to make the compensation current icj track the command current by controlling the on-off of the six switching devices S ₁ ~ S ₆ It ensures that the grid side only contains sinusoidal active components; at the same time, the DC side voltage v _dc is stabilized at the reference voltage According to Kirchhoff's law, the voltage and current equation of the three-phase active power filter is established:

p_j(j＝a,b,c)表示j相开关状态，例如当S₁闭合且S₂断开，p_a＝1；否则S₁断开S₂闭合，p_a＝0。S_j(j＝a,b,c)为开关函数，其定义如下：p _j (j=a,b,c) represents the switching state of phase j, for example, when S ₁ is closed and S ₂ is open, p _a =1; otherwise, S ₁ is open and S ₂ is closed, p _a =0. S _j (j=a,b,c) is a switching function, which is defined as follows:

系统(6)的切换平衡点x_e，为三相指令谐波电流及直流侧参考电压，记为指令谐波电流可以直接测到，直流侧参考电压为一个给定的常量。当系统(6)处在切换平衡点x_e时，则有：The switching balance point x _e of the system (6) is the three-phase command harmonic current and the reference voltage of the DC side, denoted as The command harmonic current can be directly measured, and the reference voltage of the DC side is a given constant. When the system (6) is at the switching equilibrium point x _e , then:

A_λx_e+Bu＝0 (8)A _λ x _e + Bu＝0 (8)

三相APF的稳定工作过程中，直流侧指令电压为常量，指令谐波电流则是不断发生变化，因而对系统(6)来说其切换平衡点总是处于一个过渡过程；正是这些切换平衡点的不断变化构成了三相AFP的动态补偿。During the stable operation of the three-phase APF, the command voltage on the DC side is constant, and the command harmonic current is constantly changing, so the switching balance point of the system (6) is always in a transition process; it is these switching balance points The constant change of points constitutes the dynamic compensation of the three-phase AFP.

系统Lyapnov函数设为V(ξ)＝ξ^TPξ，式中ξ＝x-x_e。如果凸组合A_λ是稳定的，则在P和Q使得The Lyapnov function of the system is set to V(ξ)=ξ ^T Pξ, where ξ=xx _e . If the convex combination A _λ is stable, then at P and Q such that

${A A}_{λ λ}^{T T} P P + + {PA PA}_{λ λ} = = - - Q Q - - - - - - ((99))$

则显然有V(ξ)＞0(x≠x_e)。那么：Then it is obvious that V(ξ)>0 (x≠x _e ). So:

$\overset{\cdot &Center Dot;}{V V} ((ξ ξ)) = = {\overset{\cdot &Center Dot;}{x x}}^{T T} Pξ Pξ + + {ξ ξ}^{T T} P P \overset{\cdot &Center Dot;}{x x} = = {22 ξ ξ}^{T T} P P (({A A}_{σ σ} x x + + Bu Bu)) - - - - - - ((1010))$

从式(10)出发，将有源电力滤波器的稳定控制问题转换为系统的镇定问题。通过设计合理的切换规则使得式(10)小于零。因此这里给出切换规则：Starting from formula (10), the stability control problem of the active power filter is transformed into the stability problem of the system. Formula (10) is less than zero by designing a reasonable switching rule. So here are the switching rules:

$σ σ ((t t)) = = arg arg \underset{i i &Element; &Element; N N}{min min} {ξ ξ}^{T T} P P (({A A}_{i i} x x + + Bu Bu)) - - - - - - ((1111))$

那么式(10)将变为：Then formula (10) will become:

$\begin{matrix} \overset{\cdot \cdot}{V V} ((ξ ξ)) = = \underset{i i &Element; &Element; N N}{min min} {ξ ξ}^{T T} [[Qξ Qξ + + 22 P P (({A A}_{i i} x x + + Bu Bu))]] - - {ξ ξ}^{T T} Qξ Qξ \\ = = \underset{i i &Element; &Element; N N}{min min} [[{ξ ξ}^{T T} (({A A}_{i i}^{T T} P P + + {PA PA}_{i i} + + Q Q)) ξ ξ + + {22 ξ ξ}^{T T} P P (({A A}_{i i} {x x}_{e e} + + Bu Bu))]] - - {ξ ξ}^{T T} Qξ Qξ \\ \underset{λ λ &Element; &Element; Λ Λ}{min min} [[{ξ ξ}^{T T} (({A A}_{λ λ}^{T T} P P + + {PA PA}_{λ λ} + + Q Q)) ξ ξ + + {22 ξ ξ}^{T T} P P (({A A}_{i i} {x x}_{e e} + + Bu Bu))]] - - {ξ ξ}^{T T} Qξ Qξ \end{matrix} - - - - - - ((1212))$

将式(8)和式(9)代入式(12)可得：Substituting formula (8) and formula (9) into formula (12) can get:

$\overset{\cdot \cdot}{V V} ((ξ ξ)) \leq \leq - - {ξ ξ}^{T T} Qξ Qξ - - - - - - ((1313))$

根据式(13)可知，切换规则σ(t)满足了系统在某个子系统运行时，Lyapnov函数导数始终为负值，即系统总是运行在为最小值的子系统；系统在切换平衡点是稳定的。According to formula (13), it can be seen that the switching rule σ(t) satisfies that when the system is running in a certain subsystem, the derivative of the Lyapnov function is always negative, that is, the system always runs in is the minimum subsystem; the system is stable at the switching equilibrium point.

图(2)为有源电力滤波器的控制框图。在截断限流补偿方式中，饱和非线性限制环节的加入，保证了有源电力滤波器安全运行不过流；但是它对原系统的影响也是相当大，甚至使系统产生极限环或新的平衡点等。因此在研究三相APF的切换控制时这种饱和非线性限制是不可忽略的。Figure (2) is the control block diagram of the active power filter. In the truncated current limiting compensation mode, the addition of the saturated nonlinear limiting link ensures the safe operation of the active power filter without overcurrent; however, it also has a considerable impact on the original system, and even causes the system to produce a limit cycle or a new equilibrium point wait. Therefore, this saturation nonlinear limitation cannot be ignored when studying the switching control of three-phase APF.

前面已设计出三相有源电力滤波器的切换规则，这里我们将利用Lyapunov稳定性定理以及凸域法，研究饱和非线性的切换稳定。当切换规则σ(t)中正定对称阵P，满足：The switching rule of the three-phase active power filter has been designed before, here we will use the Lyapunov stability theorem and the convex domain method to study the switching stability of saturated nonlinearity. When the positive definite symmetric matrix P in the switching rule σ(t) satisfies:

式中，D_n是对角线为1或0的n阶对角阵的集合，含有2ⁿ个矩阵；令D_i为D_n内一个矩阵，I为n阶单位阵。In the formula, D _n is a set of n-order diagonal matrices whose diagonals are 1 or 0, containing 2 ⁿ matrices; let D _i be a matrix in D _n , I is a unit matrix of order n.

那么含有饱和非线性限制的有源电力滤波器将是渐近稳定的。Then the active power filter with saturated nonlinear limit will be asymptotically stable.

由于(14)式中存在未知的矩阵G，不能直接进行计算，为此这里给出了一种可行的计算方法：Since there is an unknown matrix G in formula (14), it cannot be directly calculated, so a feasible calculation method is given here:

步骤(1)选择Q>0，通过下面Lyapunov等式，求解正定对称阵P：Step (1) Select Q>0, and solve the positive definite symmetric matrix P through the following Lyapunov equation:

${A A}_{λ λ}^{T T} P P + + {PA PA}_{λ λ} = = - - Q Q$

如有解，那么置循环次数k＝0，转入下一步。If there is a solution, then set the number of cycles k=0 and go to the next step.

步骤(2)使用已得的P，求解如下不等式最优问题，得到矩阵G和α，其中α为变参量：Step (2) Use the obtained P to solve the following inequality optimization problem, and obtain the matrix G and α, where α is a variable parameter:

$\underset{G G}{inf inf} α α$

$s the s . . t t . . {(({D D.}_{i i} {A A}_{λ λ} + + {D D.}_{i i}^{- -} G G))}^{T T} P P + + P P (({D D.}_{i i} {A A}_{λ λ} + + {D D.}_{i i}^{- -} G G)) < < αP αP - - - - - - ((1515))$

h_iGy_i<0h _i Gy _i <0

i＝1,2,…,ni=1,2,...,n

其中， in,

如果α＜0或者对比上一次所得的α，即α_k，如果有存在k>0和α＞α_k，转入步骤(4)；否则置循环次数k＝k+1，α_k＝α，其中α_k为变参量第K次计算值，转入下一步。If α<0 or compare the α obtained last time, i.e. α _k , if there is k>0 and α>α _k , turn to step (4); otherwise set the number of cycles k=k+1, α _k =α, Among them, α _k is the Kth calculated value of the variable parameter, and it is transferred to the next step.

步骤(3)使用上一步的G，求解如下不等式最优问题得到P和α：Step (3) Use the G in the previous step to solve the following inequality optimization problem to obtain P and α:

$\underset{G G}{inf inf} α α$

$s the s . . t t . . {(({D D.}_{i i} {A A}_{λ λ} + + {D D.}_{i i}^{- -} G G))}^{T T} P P + + P P (({D D.}_{i i} {A A}_{λ λ} + + {D D.}_{i i}^{- -} G G)) < < αP αP$

i＝1,2,...,ni=1,2,...,n

P>0P>0

如果α＜0或α＞α_k或者k_max满足时，转入步骤(4)；否则置k＝k+1，α＝α_k，返回步骤(2)。If α<0 or α>α _k or k _max is satisfied, turn to step (4); otherwise, set k=k+1, α=α _k , and return to step (2).

步骤(4)如果α＜0，那么系统大范围渐近稳定于平衡点，否则不能判断系统是否稳定。因此可以回到步骤(1)通过改变Q值，重新计算。Step (4) If α < 0, then the system is asymptotically stable at the equilibrium point in a large range, otherwise it cannot be judged whether the system is stable. So you can go back to step (1) and recalculate by changing the Q value.

附算例：Additional calculation example:

系统电压为327V/50Hz，R＝0.5Ω，L＝10mH，C＝2200μF，直流侧参考电压APF的额定电流I_max＝14A。控制系统的总体结构如图2所示。The system voltage is 327V/50Hz, R＝0.5Ω, L＝10mH, C＝2200μF, DC side reference voltage The rated current I _max of the APF is 14A. The overall structure of the control system is shown in Figure 2.

初始APF对整流桥进行谐波补偿，在0.5s时负载发生突变。APF的控制策略分别采用不考虑饱和非线性及考虑饱和非线性两种切换方式。设前者为方式1，后者为方式2。通过仿真，对这两种控制策略的补偿效果进行比较。The initial APF performs harmonic compensation to the rectifier bridge, and the load changes suddenly at 0.5s. The control strategy of APF adopts two switching modes without considering saturated nonlinearity and considering saturated nonlinearity respectively. Let the former be mode 1 and the latter be mode 2. Through simulation, the compensation effects of the two control strategies are compared.

负载突变后电网谐波电流增大，由于APF的自身容量的限制，需要根据额定容量对参考谐波进行抑制。图3中，i_L1,i_L2及i_L分别为整流桥电流，加入的突变电流以及两者之和。图4和图5是在方式1下，APF的谐波参考电流、输出补偿电流、电网电流和直流侧电压波形。从图4和图5可知，饱和非线性限制环节的存在导致原本稳定的系统失稳，此时APF不仅不能补偿谐波电流，甚至对自身和电网造成巨大危害。The grid harmonic current increases after a sudden load change. Due to the limitation of the APF's own capacity, it is necessary to suppress the reference harmonics according to the rated capacity. In Fig. 3, i _L1 , i _L2 and i _L are respectively the rectifier bridge current, the abrupt current added and the sum of the two. Figure 4 and Figure 5 are the harmonic reference current, output compensation current, grid current and DC side voltage waveforms of the APF in Mode 1. It can be seen from Figure 4 and Figure 5 that the existence of the saturated nonlinear limiting link leads to the instability of the originally stable system. At this time, the APF not only cannot compensate the harmonic current, but even causes great harm to itself and the power grid.

图6和图7是加入状态饱和系统稳定性分析后所得的仿真波形。其中，和分别为原参考电流和经限值处理后的参考电流，i_c为APF实际输出补偿电流，THD_i_L和THD_i_s为负载侧和电网侧的电流总畸变率，i_s为电网侧电流，v_dc为直流侧电压波形。比较图4和图6，以及图5和图7，通过采用基于状态饱和的切换控制策略，三相APF可以有效的补偿电网中谐波电流。Figure 6 and Figure 7 are the simulation waveforms obtained after adding state saturation system stability analysis. in, and are the original reference current and the reference current after limit value processing, i _c is the actual output compensation current of APF, THD_i _L and THD_i _s are the total current distortion rate of load side and grid side, i _s is the grid side current, v _dc is the DC side voltage waveform. Comparing Figure 4 and Figure 6, and Figure 5 and Figure 7, by adopting the switching control strategy based on state saturation, the three-phase APF can effectively compensate the harmonic current in the power grid.

Claims

1. A saturation switching control method of a three-phase parallel active power filter is characterized by comprising the following steps:

step 1) firstly analyzing the topological structure of each working mode of the three-phase parallel active power filter, and establishing a switching system model of the three-phase active power filter according to actual circuit parametersWherein A is_σ(t)Coefficient matrix of active power filter working mode, B is input matrix and is constantThe two matrixes are determined by actual circuit parameters, x is a state vector and consists of output compensation current and direct-current side voltage of a three-phase parallel active power filter, and u is an input vector and consists of system power supply voltage;

step 2) switching system model for three-phase active power filterDesigning a switching rule sigma (t) of the model to ensure that the system is gradually stabilized at a switching balance point, namely realizing the uniform control of voltage stabilization and current tracking of the three-phase active power filter; firstly, a public Lyapunov function V (xi) ═ xi of the system is calculated according to a method of a linear system theory^TPξ,(ξ＝x-x_e)，x_eFor switching balance points of the system, i.e. solvingWherein Q is a matrix of units and Q is a unit,according to the Lyapunov function switching ruleWherein arg represents a subscript;

A_λis a Hurwitz matrix and is a Hurwitz matrix,

λ_i∈(0,1)，

and 3) controlling the on-off of a switch of the three-phase active power filter according to the control rule obtained by the method, so as to realize the compensation of the harmonic current of the power grid.

2. The saturation switching control method of the three-phase parallel active power filter as claimed in claim 1, wherein the specific steps of the step 1) are:

according to the operating principle of three-phase APF, i.e. by controlling 6 switching devices S₁～S₆Make and break of (d) to make the compensating current i_cjTracking command currentThe power grid side is ensured to only contain sine active components; simultaneously make the DC side capacitor voltage v_dcStabilized at a reference voltage

Establishing a voltage-current equation of the three-phase active power filter based on kirchhoff's law:

\{\begin{matrix} e_{a} = L \frac{{di}_{ca}}{dt} + {Ri}_{ca} + U_{an} + U_{nN} \\ e_{b} = L \frac{{di}_{cb}}{dt} + {Ri}_{cb} + U_{bn} + U_{nN} \\ e_{c} = L \frac{{di}_{cc}}{dt} + {Ri}_{cc} + U_{cn} + U_{nN} \end{matrix} - - - (1)

p_j(j ═ a, b, c) represents the j-phase switching state, and L is the three-phase filter inductance; c is a direct current side capacitor; r is an inductance equivalent resistance; e.g. of the type_j(j ═ a, b, c) is the system supply voltage; v. of_dcIs the DC side capacitor voltage; i.e. i_cjA compensation current for a three-phase APF; order toRepresenting harmonic and reactive components in the load current; u shape_jn(j ═ a, b, c) is the voltage between j phase and n point; u shape_nNIs the voltage between N and the neutral point N;

definition of S_j(j ═ a, b, c) is a switching function, defined as follows:

then equations of state (1) and (3) are written as:

[\begin{matrix} \frac{{di}_{ca}}{dt} \\ \frac{{di}_{cb}}{dt} \\ \frac{{di}_{cc}}{dt} \\ \frac{{dv}_{dc}}{dt} \end{matrix}] = [\begin{matrix} - \frac{R}{L} & 0 & 0 & - \frac{S_{a}}{L} \\ 0 & - \frac{R}{L} & 0 & - \frac{S_{b}}{L} \\ 0 & 0 & - \frac{R}{L} & - \frac{S_{c}}{L} \\ \frac{p_{a}}{C} & \frac{p_{b}}{C} & \frac{p_{c}}{C} & 0 \end{matrix}] = [\begin{matrix} i_{ca} \\ i_{cb} \\ i_{cc} \\ v_{dc} \end{matrix}] + [\begin{matrix} \frac{e_{a}}{L} \\ \frac{e_{b}}{L} \\ \frac{e_{c}}{L} \\ 0 \end{matrix}] - - - (5)

for three-phase active power filter, the upper and lower bridge arms of a bridge are not turned off, and p is used_a，p_b，p_cCorresponding to 6 switch combinations, namely: 001. 010, 011, 100, 101, and 110, defining the combinations corresponding to 6 switching modes σ (t) ∈ N ═ 1,2, …, and 6, respectively; according to the topological structure of each working mode, the state equation of formula (5) is written as:

in the formula, the state vector x of APF is ═ i_ca,i_cb,i_cc,v_dc]^TAnd the input vector u ═ e_a,e_b,e_c,0]^T(ii) a A and B are coefficient matrixes of APF working modes; wherein A is_σ(t)∈{A₁,A₂,A₃,A₄,A₅,A₆For the input matrix B, the input matrix is not influenced by the switching process and is kept unchanged; for each switch combination, getWherein:

3. the saturation switching control method of the three-phase parallel active power filter as claimed in claim 2, wherein the specific steps of the step 2) are:

handoverBalance point x_eThree-phase command harmonic current and DC-side reference voltage, are recorded asThe harmonic current is directly detected, and the reference voltage at the direct current side is a given constant; according to 6 working modes of the three-phase active power filter, A is provided_σ(t)∈{A₁,A₂,A₃,A₄,A₅,A₆}; defining convex combinations of subsystems asWherein λ_i∈Λ，If convex combination A_λIs a Hurwitz matrix, i.e. A_λIs stable, then P and Q are present such that,

this is true.

4. The saturation switching control method of three-phase parallel active power filter as claimed in claim 1, wherein when the rule of step (2) is applied in practical engineering, the active power filter operates stably in the cut-off current-limiting strategy, in order to improve the effective control capability and overload capability of the active power filter to the output compensation current, the cut-off current-limiting protection strategy is usually adopted to make it operate safely, but there is a saturation non-linear restriction link in the strategy, which will destabilize the original stable system, therefore according to the Lyapunov stability theory and the convex domain method, only when the current is in the cut-off current-limiting protection strategy, the current is only stableThe positive definite symmetric array P satisfies the inequality:

<math> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>D</mi> <mi>i</mi> </msub> <msub> <mi>A</mi> <mi>λ</mi> </msub> <mo>+</mo> <msubsup> <mi>D</mi> <mi>i</mi> <mo>-</mo> </msubsup> <mi>G</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mi>P</mi> <mo>+</mo> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>D</mi> <mi>i</mi> </msub> <msub> <mi>A</mi> <mi>λ</mi> </msub> <mo>+</mo> <msubsup> <mi>D</mi> <mi>i</mi> <mo>-</mo> </msubsup> <mi>G</mi> <mo>)</mo> </mrow> <mo><</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>D</mi> <mi>i</mi> </msub> <mo>&Element;</mo> <msub> <mi>D</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1,2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow> </math>

when the active power filter is in the cutoff current-limiting strategy, the active power filter works stably;

wherein D is_nIs a set of n-order diagonal arrays with 1 or 0 diagonal, containing 2ⁿA matrix; let D_iIs D_nThe matrix is a matrix of a plurality of matrices,i is an n-order unit array; g is an unknown matrix.