CN103066601A - Hybrid active direct current filter control method based on self-adaptive linear neurons - Google Patents

Abstract

The invention discloses a hybrid active direct current filter control method based on self-adaptive linear neuron networks. The hybrid active direct current filter control method includes a first step of establishing a model of a hybrid active direct current filter as a controlled object, a second step of using the three linear neuron networks respectively as a recognizer, a controller and a predictor, and establishing a control system for the controlled object, a third step of using the recognizer to track the dynamic characteristics of the controlled object, and enabling the dynamic characteristics to be reflected on weighted vectors of the linear neuron networks, a fourth step of enabling the controller to do timely adjustment according to the dynamic changes of network parameters of the third step, and providing excellent control quantity, and a fifth step of enabling the predictor to compensate inherent hysteresis quality of the control system through a learning principle. Through the learning principle of the self-adaptive linear neuron networks to track the dynamic characteristics of the controlled object in real time and adjust the parameters of the controller on line, the hybrid active direct current filter control method solves the problems of uncertainty and time-varying characteristics of the parameters of the controlled object, and enables the performance of a hybrid active direct current filter control system to reach the optimum.

Description

Mixing active dc filter control method based on adaline

Technical field

The present invention relates to a kind of HVDC mixing active dc filter based on adaline (ADALINE, Adaptive Linear Neuron) network (HADF, Hybrid Active DC Filter) control method.

Background technology

Because the shortcoming that passive filter self is intrinsic and the fast development of power electronic technology and digital signal processor (DSP) mix active dc filter and have been applied to the HVDC(high voltage direct current) filtering of DC side harmonics electric current.Mixing active dc filter and combine the large capacity of passive filter and the good advantage of active filter dynamic characteristic, is a kind of desirable filter.

Yet the filter effect that mixes active dc filter is subjected to the impact of its control strategy to a great extent.Main comb filter or the band stop filter of adopting is as its closed loop controller at present.Some academic documents have in recent years proposed other control strategy, as based on the control of interference cancellation, High Gain Feedback control and approximate inverse-system control, but they more or less all exist that stability margin is little, the shortcoming of poor robustness, complex structure and filtered spectrum narrow range.Adaptive control has solved uncertainty and the time variation of control object parameter, but amount of calculation is larger in real time.

Summary of the invention

The present invention is directed to that existing HVDC mixed DC FILTER TO CONTROL strategy stability margin is little, the shortcoming of poor robustness, complex structure, filtered spectrum narrow range and algorithm complexity, a kind of mixing active dc filter control method based on the adaline network is provided.

For achieving the above object, the present invention adopts following technical scheme:

Mixing active dc filter control method based on adaline may further comprise the steps:

(1) set up the Mathematical Modeling of HVDC mixing active dc filter, and with this model as control object;

(2) set up three linear neuron networks, respectively as identifier, controller and fallout predictor, utilize identifier, controller and fallout predictor to set up the control system of control object;

(3) fallout predictor in the control system is by the Widrow-Holf learning rule, according to former constantly (k-1) T in the control system _s, (k-2) T _s, (k-3) T _s... (k-n) T _s, (T _sBe the sampling period, k is sampling instant, and n is natural number) reference quantity to (k+1) T constantly in future _sReference quantity predict, and premeasuring is flowed to controller, the hysteresis quality that compensation control system is intrinsic;

(4) identifier in the control system, its input comprises output and the control object output of controller, its output compares with the actual output of control object, result relatively inputs to the Widrow-Holf learning rule, identifier is adjusted its weight vectors according to the Widrow-Holf learning method, and the dynamic characteristic of control object is followed the tracks of;

(5) controller in the control system, its input comprise control object output and predict its output, adjust its weight vectors according to the Widrow-Holf learning rule of step (4), provide the Optimal Control amount, export to control object.

Linear neuron network in the described step (2) refers to:

Input vector is:

X＝[x ₁,x ₂，…,x _n] ^T

In the formula, x _i(i=1,2 ... n), n is natural number, is each element of linear neuron network input vector;

Weight vectors is:

Ω＝[ω ₁,ω ₂,…,ω _n] ^T

In the formula, ω _i(i=1,2 ... n), n is natural number, is each element of linear neuron network weight vectors;

Network is output as:

$y={\mathrm{\Ω}}^{T}X=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\ω}}_i{x}_i$

The described identifier of step (2),

Input vector is:

X(k)＝[-y(k),…,-y(k-n+1),u(k),…,u(k-m+1)] ^T

In the formula, u (k) is k controlled quentity controlled variable constantly, i.e. the constantly output of controller, and y (k) is the constantly output variable of control object of k, m, n are respectively control object transfer function molecule and denominator exponent number;

Weight vectors is:

W(k)＝[w ₁(k),…w _n(k),w _n+1(k),…w _n+m(k)] ^T

In the formula, w _i(k) (i=1,2 ... n+m) be constantly each element of identifier weight vectors of k, m, n are respectively control object transfer function molecule and denominator exponent number;

Identifier is output as:

$\hat{y}\left(k\right)=W^T(k-1)X(k-1)$

The identifier weight vectors upgrades and adopts the Widrow-Holf rule:

$W(k+1)=W\left(k\right)+\frac{{\mathrm{\αe}}_i(k+1)X\left(k\right)}{\mathrm{\ϵ}+X^T\left(k\right)X\left(k\right)}$

In the formula, α ∈ (0,2) is learning rate; ε prevents that X (k)=0 o'clock divisor from being zero a little positive number of establishing; e _i(k+1) expression k+1 predicated error constantly, the iterative initial value W (0) of weight vectors determines according to the actual measurement parameter of control object.

The described controller of step (2),

Input vector is:

$C*\left(k\right)={[y_d(k+1),\hat{y}\left(k\right),y(k-1),\·\·\·,y(k-n+1),-u(k-1),\·\·\·,u(k-m+1)]}^T$

Y in the formula _d(k+1) be to the constantly prediction of reference quantity of k+1, can obtain by fallout predictor, wherein m, n are respectively control object transfer function molecule and denominator exponent number;

The weight vectors of controller is

$V\left(k\right)=w_{n+1}^{-1}\left(k\right){[1,w_1\left(k\right),\·\·\·w_n\left(k\right),w_{n+2}\left(k\right),\·\·\·w_{n+m}\left(k\right)]}^T$

M, n are respectively control object transfer function molecule and denominator exponent number in the formula;

Controller is output as

u(k)＝V ^T(k-1)C*(k)；

The input vector of step (5) fallout predictor is:

P(k)＝[y _r(k-1),y _r(k-2),…,y _r(k-l)] ^T

Y in the formula _r(k) be constantly reference quantity, the i.e. negative value of converter side harmonic current sampled value of k; L is length of window, namely to k constantly before the number of samples of harmonic wave

Weight vectors is:

Q(k)＝[q ₁(k),q ₂(k),…,q _l(k)] ^T

Q in the formula _i(k) (i=1,2 ... each element of fallout predictor weight vectors when .l) being k.The premeasuring of fallout predictor output is:

y _d(k+1)＝Q ^T(k)P(k)

The fallout predictor weights adopt the Widrow-Holf rule to upgrade:

$Q(k+1)=Q\left(k\right)+\frac{{\mathrm{\αe}}_p(k+1)P\left(k\right)}{\mathrm{\ϵ}+P^T\left(k\right)P\left(k\right)}$

The invention has the beneficial effects as follows: set up the HVDC dc active power filter control strategy based on the adaline network, this control strategy is a kind of adaptive open-loop Tracking Control Strategy, learning algorithm by the adaline network carries out real-time tracking to the dynamic characteristic of control object, the online controller parameter of adjusting, overcome uncertainty and the time variation of control object parameter, and according to the harmonic current sampled value in the former moment harmonic current that will constantly occur future is predicted, the hysteresis quality that exists with compensation control system self makes the performance of HVDC mixing active dc filter control system reach optimum to the adverse effect that control system causes; There is not stability problem common in the Closed-loop Control Strategy in this control strategy yet simultaneously; Since harmonic current numerical value larger converter side and filter branch side are carried out current sample, the interference problem when this has just been avoided weak signal sampling (line side), and it is little to have amount of calculation, is easy to the advantage of Digital Implementation.

Description of drawings

Fig. 1 is schematic diagram of the present invention;

Fig. 2 is single linear neuron network configuration;

Fig. 3 is HVDC mixing active dc filter topological structure;

Fig. 4 is HVDC mixing active dc filter equivalent electric circuit;

Fig. 5 is the actual output of reference input, the control object of control system and departure.

Embodiment

The present invention will be further described below in conjunction with accompanying drawing and embodiment.

Based on the HVDC mixing active dc filter control system structure of linear neuron network as shown in Figure 1.Y among the figure _r(k-1) be k-1 harmonic current sampled value constantly; U (k) is k controlled quentity controlled variable constantly; y _r(k+1), y _d(k+1), ξ (k+1), y (k+1),

e _p(k+1) and e _i(k+1) be respectively k+1 reference quantity (i.e. the negative value of this moment converter side harmonic current sampled value), premeasuring, control object disturbance, the actual output of control object, identifier output, predicated error and Identification Errors constantly.Single linear neuron network configuration as shown in Figure 2, input vector is among the figure

X＝[x ₁,x ₂，…,x _n] ^T

In the formula, x _i(i=1,2 ... n, n are natural number) be each element of input vector; Weight vectors is

Ω＝[ω ₁,ω ₂,…,ω _n] ^T

In the formula, ω _i(i=1,2 ... n, n are natural number) be each element of weight vectors; Network is output as

$y={\mathrm{\Ω}}^{T}X=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\ω}}_i{x}_i$

For identifier and fallout predictor, its learning rule adopts the Widrow-Holf rule

$W(k+1)=W\left(k\right)+\frac{\mathrm{\αe}(k+1)X\left(k\right)}{\mathrm{\ϵ}+X^T\left(k\right)X\left(k\right)}$

In the formula, α _∈(0,2) is learning rate; ε prevents that X (k)=0 o'clock divisor from being zero a little positive number of establishing; The iterative initial value W (0) of weight vectors determines according to the actual measurement parameter of control object.

The below introduces the process of setting up and identifier, controller and the fallout predictor of the Mathematical Modeling of control object in detail.

1, control object Mathematical Modeling

Passive filter is double-tuned filter commonly used in the HVDC system, by resistance R ₁, R ₂, capacitor C ₁, C ₂And inductance L ₁, L ₂Consist of, and ignore the loss of capacitor.Direct current hybrid active filter structure comprises that double-tuned filter, coupling transformer, voltage source inverter, DC power supply, PWM generate link and controller as shown in Figure 3, and current sensor 1 and 2 detects respectively converter harmonic current I _S(reference input) and filter output offset current I _F(actual output) is as input signal and the feedback signal of controller.Active filter adopts Tracking Control Strategy, and the control target is to make I _F=-I _SThereby, eliminate the harmonic current I in the circuit _L

Fig. 4 is HVDC dc active power filter equivalent electric circuit, U among the figure _SBe harmonic voltage source; Z _SBe the voltage source internal impedance, i.e. converter internal impedance and smoothing reactor internal impedance sum; Z _PFBe the filter branch impedance, mainly consider passive filter impedance, U herein _INVBe inverter output voltage; K is the coupling transformer no-load voltage ratio; Z _LEquivalent load impedance for Inverter Station and transmission line.According to the KCL theorem, can get

-KU _INV(s)+Z _PF(s)I _F(s)+Z _L(s)I _L(s)＝0

Again because the control target is I _L=0, so

-KU _INV(s)+Z _PF(s)I _F(s)＝0

Therefore the transfer function of control object is

$G_P\left(s\right)=\frac{I_F\left(s\right)}{U_{\mathrm{INV}}\left(s\right)}=\frac{K}{Z_{\mathrm{PF}}\left(s\right)}$

Have for double-tuned filter shown in Figure 1

$Z_{\mathrm{PF}}\left(s\right)={\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="HDA00002591513900011.png"\; state="\{\{state\}\}">R</figure-callout>}}_1+{\mathrm{<figure-callout\; id="1"\; label="sL"\; filenames="HDA00002591513900011.png"\; state="\{\{state\}\}">sL</figure-callout>}}_1+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="sC"\; filenames="HDA00002591513900011.png"\; state="\{\{state\}\}">sC</figure-callout>}}_1}+\frac{R_2+{\mathrm{sL}}_2}{s^2{L}_2{C}_2+s{R}_2{C}_2+1}$

The discrete transfer function of control object can represent with following form

$G_P\left(z^{-1}\right)=\frac{b_1{z}^{-1}+b_2{z}^{-2}+b_3{z}^{-3}+b_4{z}^{-4}}{1+a_1{z}^{-1}+a_2{z}^{-2}+a_3{z}^{-3}+a_4{z}^{-4}}$

Because double-tuned filter is the principal mode of DC side passive filtration unit, the situation when therefore double-tuned filter is adopted in main consideration.

2, based on the control strategy of adaline network

2.1 identifier design

Identifier is followed the tracks of the dynamic response of control object by learning algorithm, and its input vector is

X(k)＝[-y(k),…,-y(k-n+1),u(k),…,u(k-m+1)] ^T

M and n are respectively the exponent number of control object molecule and denominator in the formula;

Weight vectors is

W(k)＝[w ₁(k),…w _n(k),w _n+1(k),…w _n+m(k)] ^T

M and n are respectively the exponent number of control object molecule and denominator in the formula.Identifier is output as

$\hat{y}\left(k\right)=W^T(k-1)X(k-1)$

The identifier weight vectors upgrades and adopts the Widrow-Holf rule

$W(k+1)=W\left(k\right)+\frac{{\mathrm{\&alpha;e}}_i(k+1)X\left(k\right)}{\mathrm{\&epsiv;}+X^T\left(k\right)X\left(k\right)}$

In the formula, α ∈ (0,2) is learning rate; ε prevents that X (k)=0 o'clock divisor from being zero a little positive number of establishing.The iterative initial value W (0) of weight vectors can determine according to the actual measurement parameter of control object.

2.2 controller design

Controller need to utilize the identification information of control object, and its input vector is

C(k)＝[y _r(k+1),y(k),…,y(k-n+1),-u(k-1),…,-u(k-m+1)] ^T

Y in the formula _d(k+1) be to the constantly prediction of reference quantity of k+1, can obtain by fallout predictor, wherein m, n are respectively control object transfer function molecule and denominator exponent number;

Corresponding weight vectors is

$V\left(k\right)=w_{n+1}^{-1}\left(k\right){[1,w_1\left(k\right),\&CenterDot;\&CenterDot;\&CenterDot;w_n\left(k\right),w_{n+2}\left(k\right),\&CenterDot;\&CenterDot;\&CenterDot;w_{n+m}\left(k\right)]}^T$

In the formula, w _i(k) (i=1,2 ... n+m) be constantly each element of identifier weight vectors of k, m, n are respectively control object transfer function molecule and denominator exponent number;

Controller is output as

u(k)＝V ^T(k)C(k)

Be not difficult to find from following formula, because the input vector C (k) in the controller k moment and weight vectors V (k) need the sampled value y (k) of current time control object output, this causes difficulty for realization of hardware.Therefore this paper adopts the constantly output of identifier of k

Replace the actual output y (k) of control object, then input vector can be rewritten as

$C*\left(k\right)={[y_r(k+1),\hat{y}\left(k\right),y(k-1),\&CenterDot;\&CenterDot;\&CenterDot;,y(k-n+1),-u(k-1),\&CenterDot;\&CenterDot;\&CenterDot;,u(k-m+1)]}^T$

And replace V (k) with V (k-1), then the controller weight vectors changes into

u(k)＝V ^T(k-1)C*(k)

Along with the carrying out of identification process, Can be tending towards gradually y (k), V (k) can level off to V (k-1) gradually.

2.3 fallout predictor design

Comprise k+1 reference quantity y constantly among the controller k input vector C (k) constantly _r(k+1), namely need k+1 harmonic current sampled value constantly, this can not realize physically.Also need an Adaline unit according to the reference quantity of k before the moment k+1 reference quantity constantly to be predicted for this reason.This Time Controller input vector is

$C*\left(k\right)={[y_d(k+1),\hat{y}\left(k\right),y(k-1),\&CenterDot;\&CenterDot;\&CenterDot;,y(k-n+1),-u(k-1),\&CenterDot;\&CenterDot;\&CenterDot;,u(k-m+1)]}^T$

In the formula, y _d(k+1) be to k+1 moment reference quantity y _r(k+1) prediction, wherein m, n are respectively control object transfer function molecule and denominator exponent number.The input vector of fallout predictor is

P(k)＝[y _r(k-1),y _r(k-2),…,y _r(k-l)] ^T

Corresponding weight vectors is

Q(k)＝[q ₁(k),q ₂(k),…,q _l(k)] ^T

Y in the formula _r(k) be constantly reference quantity, the i.e. negative value of harmonic current sampled value of k; L is length of window, adopt shorter length of window can obtain preferably dynamic property, but precision of prediction is relatively poor; Long length of window can obtain preferably precision of prediction, but dynamic property is relatively poor.Fallout predictor output

y _d(k+1)＝Q ^T(k)P(k)

The fallout predictor weights adopt the Widrow-Holf rule to upgrade equally

$Q(k+1)=Q\left(k\right)+\frac{{\mathrm{\&alpha;e}}_p(k+1)X\left(k\right)}{\mathrm{\&epsiv;}+X^T\left(k\right)X\left(k\right)}$

Compiling Matlab program is carried out Calculation Verification to above control strategy.

For studying conveniently, " Predictive current control strategy based on ADALINE for active power filter " gets fallout predictor length of window l=5 according to document, and the weight vectors initial value is

Q(0)=[0.5,0.5,0.5,0.5,0.5] ^T

Corresponding learning algorithm parameter alpha=1.0, ε=0.001.The reference quantity of fallout predictor is taken from one 6 pulsation rectifier DC side current waveform, as shown in Figure 5.

For double-tuned filter discussed in this article, the molecule of control object transfer function and denominator are 4 rank, i.e. m=n=4.The initial value of weight vectors can draw according to the actual measurement calculation of parameter to the control object parameter.Parameter in the table 1 is as example,

Table 1 compound filter each several part parameter

The initial value of weight vectors is

W(0)=[-3.3787,4.0817,-3.3432,0.9790,0.2500,-0.6417,0.6015,-0.2025] ^T

Corresponding learning algorithm parameter alpha=1.0, ε=0.001.The disturbance that exists in the control object is that a scope is at the white noise sequence of [0.0025,0.0025].

The weight vectors of controller upgrades along with the renewal of identifier weight vectors.The output of control object and departure in this control system (difference of control object output and reference input) are as shown in Figure 5.

Described HVDC mixing active dc filter control strategy based on the adaline network is a kind of adaptive open-loop Tracking Control Strategy that is made of fallout predictor, identifier and controller, need to measure the harmonic current of converter side and filter branches side, the control target is to make the two amplitude equal phase opposite;

Described control object comprises the passive filter (such as double-tuned filter) of all forms that is applied to the HVDC DC side and the cascaded structure of single-phase electricity die mould inverter;

Described fallout predictor is an adaline network, its weight vectors upgrades by the Widrow-Holf learning rule, input vector and weight vectors dimension m+n(m, n are respectively control object transfer function molecule and denominator exponent number), weight vectors initial value Q (0), learning rate α and little positive number ε not only be confined to this paper value, can be according to concrete applicable cases setting;

Described identifier is an adaline network, its weight vectors upgrades by the Widrow-Holf learning rule, weight vectors initial value W (0), learning rate α and little positive number ε not only are confined to this paper value, can be according to concrete applicable cases setting;

Described controller is a linear neural network, and its input comprises fallout predictor output, identifier output and controller and the in the past output in the moment of control object, and weight vectors is determined by the weight vectors of identifier.

Claims (5)

1. based on the mixing active dc filter control method of adaline, it is characterized in that, may further comprise the steps:

(1) set up the Mathematical Modeling of HVDC mixing active dc filter, and with this model as control object;

(2) set up three linear neuron networks, respectively as identifier, controller and fallout predictor, utilize identifier, controller and fallout predictor to set up the control system of control object;

(3) fallout predictor in the control system is by the Widrow-Holf learning rule, according to former constantly (k-1) T in the control system _s, (k-2) T _s, (k-3) T _s... (k-n) T _sReference quantity to (k+1) T constantly in future _sReference quantity predict, and premeasuring is flowed to controller, the hysteresis quality that compensation control system is intrinsic, wherein T _sBe the sampling period, k is sampling instant, and n is natural number;

(4) identifier in the control system, its input comprises output and the control object output of controller, its output compares with the actual output of control object, result relatively inputs to the Widrow-Holf learning rule, identifier is adjusted its weight vectors according to the Widrow-Holf learning method, and the dynamic characteristic of control object is followed the tracks of;

(5) controller in the control system, its input comprise control object output and predict its output, adjust its weight vectors according to the Widrow-Holf learning rule of step (4), provide the Optimal Control amount, export to control object.

2. the mixing active dc filter control method based on adaline as claimed in claim 1, it is characterized in that: the linear neuron network in the described step (2) refers to:

Input vector is:

X＝[x ₁,x ₂，…,x _n] ^T

In the formula, x _i(i=1,2 ... n) be each element of linear neuron network input vector, n is natural number;

Weight vectors is:

Ω＝[ω ₁,ω ₂,…,ω _n] ^T

In the formula, ω _i(i=1,2 ... n) be each element of linear neuron network weight vectors, n is natural number;

Network is output as:

$y={\mathrm{\&Omega;}}^{T}X=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_i{x}_i.$

3. the mixing active dc filter control method based on adaline as claimed in claim 1 is characterized in that: the described identifier of step (2), and its input vector is:

X(k)＝[-y(k),…,-y(k-n+1),u(k),…,u(k-m+1)] ^T

In the formula, u (k) is k controlled quentity controlled variable constantly, i.e. the constantly output of controller, and y (k) is the constantly output variable of control object of k, m, n are respectively control object transfer function molecule and denominator exponent number;

Weight vectors is:

W(k)＝[w ₁(k),…w _n(k),w _n+1(k),…w _n+m(k)] ^T

In the formula, w _i(k) (i=1,2 ... n+m) be constantly each element of identifier weight vectors of k; M and n are respectively the exponent number of control object molecule and denominator in the formula;

Identifier is output as:

$\hat{y}\left(k\right)=W^T(k-1)X(k-1)$

The identifier weight vectors upgrades and adopts the Widrow-Holf rule:

$W(k+1)=W\left(k\right)+\frac{{\mathrm{\&alpha;e}}_i(k+1)X\left(k\right)}{\mathrm{\&epsiv;}+X^T\left(k\right)X\left(k\right)}$

In the formula, α ∈ (0,2) is learning rate; ε prevents that X (k)=0 o'clock divisor from being zero a little positive number of establishing; e _i(k+1) expression (k+1) predicated error constantly.

4. the mixing active dc filter control method based on adaline as claimed in claim 1 is characterized in that: the described controller of step (2),

Input vector is:

$C*\left(k\right)={[y_d(k+1),\hat{y}\left(k\right),y(k-1),\&CenterDot;\&CenterDot;\&CenterDot;,y(k-n+1),-u(k-1),\&CenterDot;\&CenterDot;\&CenterDot;,u(k-m+1)]}^T$

Y in the formula _d(k+1) be that m, n are respectively control object transfer function molecule and denominator exponent number to the constantly prediction of reference quantity of k+1;

Weight vectors is:

u(k)＝V ^T(k-1)C*(k)

Controller is output as:

u(k)＝V ^T(k)C(k)。

5. the mixing active dc filter control method based on adaline as claimed in claim 1 is characterized in that: step (5) fallout predictor,

Input vector is:

P(k)＝[y _r(k-1),y _r(k-2),…,y _r(k-l)] ^T

Y in the formula _r(k) be constantly reference quantity, the i.e. negative value of harmonic current sampled value of k; L is length of window, namely to k constantly before the number of samples of harmonic wave;

Weight vectors is:

Q(k)＝[q ₁(k),q ₂(k),…,q _l(k)] ^T

Q in the formula _i(k) (i=1,2 ... each element of fallout predictor weight vectors when .l) being k;

Fallout predictor is output as:

y _d(k+1)＝Q ^T(k)P(k)

The fallout predictor weights adopt the Widrow-Holf rule to upgrade:

$Q(k+1)=Q\left(k\right)+\frac{{\mathrm{\&alpha;e}}_p(k+1)P\left(k\right)}{\mathrm{\&epsiv;}+P^T\left(k\right)P\left(k\right)}.$

Images