Lithium battery remaining useful life prediction method based on fusion data driven model

Abstract

The application discloses a lithium battery residual service life prediction method based on a fusion data-driven model, and the method comprises the following steps: extracting the constant current charging duration, the constant voltage charging duration, the vertical slope at the corner of the constant current charging curve and the vertical slope at the corner of the constant current discharging curve as health characteristics; taking the four health characteristics as inputs and corresponding battery capacities as outputs to train a chaos sparrow-extreme learning machine model and a least square support vector regression model; obtaining the battery capacity prediction values by using the two trained models respectively, and performing weighted fusion to obtain the final lithium battery capacity prediction value; and finally obtaining the residual service life of the lithium battery in combination with the lithium battery capacity curve. The CSSA-ELM-LSSVR fusion algorithm can fully utilize the CSSA-ELM to extract the overall trend of the lithium battery degradation process, and utilize the LSSVR to obtain the local nonlinear characteristics, so that the accurate prediction of the residual service life of the lithium battery is realized, and the good robustness is also achieved.

Description

Technical Field

[0001] This invention belongs to the field of battery technology, specifically relating to a method for predicting the remaining lifespan of lithium batteries based on a fusion data-driven model. Background Technology

[0002] In recent years, lithium-ion batteries, with their outstanding advantages such as high output voltage, high energy density, good cycle performance, low self-discharge, no memory effect, and environmental friendliness, have taken on the role of core energy storage components in many fields such as new energy vehicles, smart grids, aerospace vehicles, medical devices, communications, and consumer electronics.

[0003] However, lithium-ion batteries have complex charging and discharging reactions, operate in harsh environments, and, with use, gradually age and degrade under different operating conditions (such as temperature, voltage, and charging / discharging current). Ignoring this degradation process and overloading the battery for extended periods can lead to anything from minor equipment malfunctions or damage to significant economic losses and even serious safety incidents.

[0004] The safety and reliability of lithium-ion batteries are bottlenecks hindering their rapid development. Efficient management of lithium-ion batteries is a crucial way to ensure the safe operation of electrical equipment and is currently a hot research topic in the energy field. Therefore, there is an urgent need for a method to predict the RUL (remaining useful life) of lithium batteries. Summary of the Invention

[0005] The purpose of this invention is to provide a method for predicting the remaining lifespan of lithium batteries based on a fusion data-driven model, so as to accurately predict the remaining lifespan of lithium batteries.

[0006] To achieve the above objectives, the technical solution of the present invention is as follows:

[0007] A method for predicting the remaining lifespan of lithium batteries based on a fusion data-driven model is proposed. This method employs a fusion algorithm of CSSA-ELM and LSSVR based on health feature extraction to achieve accurate prediction of the remaining lifespan of lithium batteries. The method includes the following steps:

[0008] The process involves acquiring lithium battery capacity data and extracting features based on constant current-constant voltage (CC-CV) charging voltage curves and constant current (CC) discharging voltage curves. Specifically, this includes: acquiring battery capacity data from NASA and MIT battery datasets and plotting capacity curves, CC-CV charging voltage curves, and CC discharging voltage curves; the lithium battery capacity curve is a battery capacity-cycle period curve; extracting constant current charging duration HI1, constant voltage charging duration HI2, and vertical slope HI3 at the corner of the constant current charging curve as health indicators from the CC-CV charging voltage curve; and extracting the vertical slope HI4 at the corner of the constant current discharging curve as a health indicator from the CC discharging voltage curve.

[0009] Offline training of the battery degradation process was performed using Chaotic Sparrow Algorithm-Extreme Learning Machine (CSSA-ELM) and Least Squares Support Vector Regression (LSSVR) algorithms, respectively. Specifically, for the CSSA-ELM algorithm, CSSA was used to optimize the initial weights ω and bias b of the ELM, the fitness function was designed based on the mean squared error (MSE) of the training set, and the ELM was constructed and trained using the obtained optimal parameters. For the LSSVR algorithm, health indicators and capacity sequences were used as input and output to train the LSSVR model.

[0010] The remaining lifespan of a lithium battery is predicted based on the CSSA-ELM-LSSVR fusion algorithm. Specifically, a data fusion method is used to combine the advantages of ELM and LSSVR, and the weights of the predicted values ​​of ELM and LSSVR are reasonably allocated. Based on the CSSA-ELM-LSSVR fusion algorithm, the remaining lifespan of the lithium battery is predicted.

[0011] To verify the effectiveness of the prediction model, error analysis is performed on the RUL prediction. The RUL prediction error, root mean square error, and coefficient of determination can be used to evaluate the model performance.

[0012] Furthermore, the method also includes:

[0013] After feature extraction of the CC-CV charging voltage curve and CC discharging voltage curve, the Pearson correlation coefficient (PCC) was used to quantitatively analyze the correlation between health indicators (HIs) and capacity.

[0014] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0015] The lithium battery remaining life prediction method based on the fusion data-driven model of the present invention adopts the CSSA-ELM-LSSVR fusion algorithm, which can make full use of CSSA-ELM to extract the overall trend of lithium battery degradation process and use LSSVR to obtain local nonlinear features, thereby achieving accurate prediction of the remaining life of lithium battery and having good robustness.

[0016] In addition, this invention extracts four health indicators: constant current charging duration, constant voltage charging duration, vertical slope at the corner of the constant current charging curve, and vertical slope at the corner of the constant current discharging curve, which can more accurately predict the remaining service life of lithium batteries. Attached Figure Description

[0017] Figure 1 This is a framework diagram of the lithium-ion battery remaining lifespan prediction method provided in the embodiments of the present invention;

[0018] Figure 2 These are the charging voltage curves of the NASA dataset B0005 group of batteries under different cycle periods used in the embodiments of this invention;

[0019] Figure 3 These are the charging voltage curves of MIT14 groups of batteries under different cycle periods from the MIT dataset used in the embodiments of the present invention.

[0020] Figure 4 These are the discharge voltage curves of the NASA dataset B0005 group of batteries under different cycle periods used in the embodiments of this invention;

[0021] Figure 5 These are the discharge voltage curves of MIT14 groups of batteries under different cycle periods from the MIT dataset used in the embodiments of the present invention. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0023] The lithium battery remaining life prediction method based on a fusion data-driven model according to embodiments of the present invention, such as... Figure 1 As shown, it includes:

[0024] Step 1: Obtain the capacity data of the lithium battery and extract features based on the CC-CV charging voltage curve and CC discharging voltage curve;

[0025] Step 2: The battery degradation process is trained offline based on the CSSA-ELM and LSSVR algorithms respectively to obtain the optimal training model;

[0026] Step 3: Predict the remaining lifespan of the lithium battery based on the CSSA-ELM-LSSVR fusion algorithm;

[0027] Step 4: To verify the effectiveness of the prediction model, an error analysis is performed on the RUL prediction.

[0028] Specifically, in Step 1, the obtained NASA and MIT datasets were first filtered, and three battery groups (B0005, B0006, and B0018) from the NASA dataset and the CH14 batteries from the "2018-02-20" batch from the MIT dataset were randomly selected as experimental datasets. Four health indicators were extracted: constant current charging time, constant voltage charging time, vertical slope at the corner of the constant current charging curve, and vertical slope at the corner of the constant current discharging curve. 60% of the data sequences were randomly selected as the training set, and the remainder as the test set.

[0029] In Step 2, this embodiment of the invention employs the CSSA-ELM algorithm and the LSSVR algorithm for single-model training, respectively. It is understood that ELM is a learning algorithm for a single hidden layer feedforward neural network, characterized by its simplicity, strong learning ability, and widespread application in health status prediction. CSSA is a novel swarm intelligence optimization algorithm inspired by sparrow foraging and anti-predation behavior, which avoids getting trapped in local optima and features high search accuracy, fast convergence speed, good stability, and strong robustness. Since the weights and biases in ELM are randomly generated, this embodiment uses CSSA to optimize the initial weights and biases of ELM. LSSVR, as a classic kernel-based learning method, is widely used in RUL prediction due to its low computational cost and strong generalization performance.

[0030] In Step 3, from the perspective of learning efficiency, ELM has the advantages of simple structure, fast learning speed, and strong generalization ability, and can fully explore the overall degradation trend of the degradation process. Meanwhile, the LSSVR algorithm has the advantages of strong nonlinear mapping and strong small-sample learning ability, and can extract the local nonlinear characteristics of the degradation process. However, single-data-driven prediction methods generally suffer from poor robustness. Therefore, this embodiment of the invention uses a fusion data-driven model method, combining the advantages of the two algorithms in Step 2, and reasonably allocating the weights of the ELM and LSSVR prediction values ​​to provide a new CSSA-ELM-LSSVR fusion algorithm for predicting the remaining lifespan of lithium batteries.

[0031] Finally, in Step 4, to verify the effectiveness of the model, this embodiment of the invention uses the RUL prediction error E. rul Root mean square error (RMSE) and coefficient of determination (R²) 2 Evaluate the model performance.

[0032] The lithium-ion battery remaining life prediction method based on the CSSA-ELM and LSSVR fusion algorithm for health feature extraction provided in this invention uses the CSSA-ELM-LSSVR fusion algorithm to achieve accurate remaining life prediction and has good robustness.

[0033] Based on the above embodiments, feature extraction is performed according to the CC-CV charging voltage curve and the CC discharging voltage curve, including:

[0034] Based on the NASA battery dataset and the MIT battery dataset, obtain battery capacity data and plot capacity curves, as well as CC-CV charging voltage curves and CC discharging voltage curves.

[0035] Based on the CC-CV charging voltage curve, the constant current charging duration HI1, constant voltage charging duration HI2, and vertical slope HI3 at the corner of the constant current charging curve are extracted as health indicators.

[0036] The vertical slope HI4 at the corner of the constant current discharge curve is extracted from the CC discharge voltage curve as a health indicator.

[0037] Specifically, in the simulation experiment, this embodiment of the invention selects three sets of battery data (B0005, B0006, and B0018) from NASA (60% of the data is randomly used as the training set, and the remainder as the test set) and CH14 battery data from MIT (60% of the data is randomly used as the training set, and the remainder as the test set) for simulation experiments. Taking B0005 battery and MIT14 battery as examples, CC-CV charging voltage curves and CC discharging voltage curves for four different cycle periods (40, 80, 120, 160) and four different cycle periods (100, 200, 300, 400) are extracted respectively. Figures 2 to 5 As shown.

[0038] Depend on Figure 2 , Figure 3 It can be seen that as the number of cycles increases, the charging voltage curve shifts to the left, the charging time in CC mode gradually decreases, the charging duration in CV mode shows an increasing trend, and the vertical slope at the corner of the curve in CC charging mode gradually increases. Therefore, in this embodiment of the invention, constant current charging duration HI1, constant voltage charging duration HI2, and vertical slope at the corner of the constant current charging curve HI3 are selected as health indicators.

[0039] Depend on Figure 4 , Figure 5 It can be seen that as the number of cycles increases, the battery voltage drops faster and faster. At the end of the CC discharge mode, the slope of the discharge curve tends to stabilize. As the battery ages, the vertical slope of the CC discharge curve corner decreases. Therefore, in this embodiment of the invention, the vertical slope HI4 at the corner of the constant current discharge curve is selected as a health indicator.

[0040] Specifically, in the simulation experiment, this embodiment of the invention randomly selects 60% of the data sequences as the training set, and the remainder as the test set. A prediction starting point T is set, and then the original data interval is converted to [0, 1] using the min-max normalization method. The normalization formula is as follows:

[0041]

[0042] Among them, y i and y norm These represent the original data and the normalized data, respectively. max and y min These are the maximum and minimum values ​​in the original data.

[0043] Based on the above embodiments, the method further includes:

[0044] After feature extraction from the CC-CV charging and CC discharging curves, the Pearson correlation coefficient (PCC) was used to quantitatively analyze the correlation between health indicators (HIs) and capacity. The formula for calculating PCC is as follows:

[0045]

[0046] Where x i , Q represents the extracted health indicator sequence and the mean of that sequence, respectively. i , Here, represents the mean of the extracted volume sequence and , respectively; n represents the dimension of the extracted sequence; the numerator of PCC is the covariance between health indicators and volume, and the denominator is the standard deviation of health indicators and the standard deviation of volume. PCC values ​​range from -1 to 1, where |PCC| closer to 1 indicates a better correlation between health indicators and volume, and 0 indicates no correlation between the two.

[0047] Based on the above embodiments, offline training is performed on the battery degradation process using CSSA-ELM and LSSVR algorithms, respectively, including:

[0048] For the CSSA-ELM algorithm, CSSA is used to optimize the initial weights ω and biases b of the ELM. The fitness function is designed according to the mean squared error (MSE) of the training set. The ELM is constructed and trained using the obtained optimal parameters to obtain the optimal CSSA-ELM model.

[0049] For the LSSVR algorithm, health indicators and capacity sequences are used as input and output to train the LSSVR model to obtain the optimal LSSVR model.

[0050] As can be seen from the above embodiments, the methods provided in the embodiments of the present invention use the CSSA-ELM algorithm and the LSSVR algorithm respectively for single-model prediction.

[0051] 1. CSSA-ELM

[0052] ELM is a learning algorithm for a single-hidden-layer feedforward neural network, consisting of three layers: an input layer, a hidden layer, and an output layer. Assume there are N samples (x... i , t i ), where x i =[x i1 x i2 , ..., x in ] T ∈R n , t i =[t i1 , t i2 , ..., t im ] T ∈R mm Represent the dataset as The ELM algorithm is represented as follows:

[0053]

[0054] Where l is the number of neurons in the hidden layer, n is the number of training samples, and x i =[x i1 x i2 , ..., x in ] T ∈R n For input, w i =[w i1 w i2 ,…,w it ] T β represents the connection weights from the input layer to the hidden layer. i =[β i1 ,β i2 , ..., β im ] T b represents the connection weight from the hidden layer to the output layer. i Let g(·) be the bias of the i-th hidden layer unit, and g(·) represent the activation function of the hidden layer.

[0055] Equation (3) can be simplified as follows:

[0056] T=Gβ (4)

[0057] in

[0058] Randomly generate weights w i and bias b i Determine the output weights:

[0059]

[0060] Among them G + =(G T G) -1 G T , where represents the generalized inverse matrix of matrix G.

[0061] Since the weights and biases in ELM are randomly generated, this embodiment of the invention incorporates the Chaotic Sparrow Algorithm (CSSA) to optimize the weights and biases of ELM. First, a chaotic map is used to initialize the population positions. Then, to avoid a decrease in population diversity during iteration, a Gaussian mutation operator is used to enhance local search capabilities. The steps of CSSA are as follows:

[0062] 1) Tent mapping:

[0063] The iteration of the Tent mapping can be described as follows:

[0064]

[0065] Where z(k)∈(0,1) is the system value of the kth iteration, and μ∈(0,2) is the chaotic control parameter of the Tent mapping function.

[0066] Based on the Tent mappings described above, CSSA initialization is achieved in the following way:

[0067] First, take the i-th sparrow in the j-th dimension. ij (i = 1, 2, ..., N; j = 1, 2, ..., D) are normalized to z using equation (6). ij (0):

[0068]

[0069] Where lb and ub are respectively z ij The lower and upper bounds.

[0070] Then, z is obtained using the Tent mapping. ij (k+1) is as follows:

[0071]

[0072] Using equation (9) to z ij (k+1) is transformed from the original domain [0, 1] into a new z. ij :

[0073] z ij =lb+z ij (k+1)(ub-lb) (9)

[0074] 2) Gaussian mutation:

[0075] To avoid getting trapped in local optima and maintain the diversity of the sparrow population during iteration, a Gaussian mutation operator is introduced to enhance local search capabilities. The Gaussian mutation formula is as follows:

[0076] mutation(z)=z(1+rand(1,D)) (10)

[0077] Based on the above, this embodiment of the invention incorporates CSSA to optimize the initial weights ω and biases b in the ELM algorithm during single-model prediction. Its fitness function is designed based on the mean squared error (MSE) of the training set:

[0078] fitness = arg min(MSE) train_set (11)

[0079] in y i and Let MSE represent the actual value and predicted value of sample i, respectively, where N represents the total number of samples. train_set The smaller the value, the stronger the correlation between the ELM output and the original capacity.

[0080] Specifically, the CSSA-ELM algorithm includes the following implementation steps:

[0081] (1) Initialization. Initialize the population size N and the number of discoverers P. N Number of scouts R N The dimension D of the objective function, the upper bound ub and lower bound lb of the initial ω and b, the maximum number of iterations T, and the solution accuracy ε.

[0082] (2) Tent mapping and transformation. Tent mapping is performed and the initial value x is transformed using equations (7)-(9). i =(x i，ω x i，b (i = 1, 2, ..., N).

[0083] (3) Train the ELM model. According to equations (3)-(5), use ω and b from equation (9) as initialization parameters to train the ELM model.

[0084] (4) Calculate the fitness of each sparrow using equation (11) and select the current optimal fitness f. best and its corresponding position and the current worst fitness f worst and its corresponding position Then, update the positions of the explorer and follower according to equations (12) and (13).

[0085]

[0086]

[0087] in This represents the position of the i-th sparrow in the j-th dimension at the current iteration number t. This is the optimal position for the current discoverer. This represents the worst position in the population. α∈(0,1) are random numbers, and Q is a random number following a standard normal distribution. L is a 1×D matrix where each element is 1. A is a 1×D matrix where each element is randomly assigned a value of 1 or -1. + =A T (AA T ) -1 R∈[0,1] and ST∈[0.5,1] are the warning value and the safety threshold, respectively. When R<ST, the scout sparrow can search extensively, guiding the population to achieve higher fitness, and there are no predators or other dangers; when R≥ST, the scout sparrow discovers a predator, the population immediately takes anti-predation actions, and all sparrows quickly migrate to other safe areas.

[0088] (5) Randomly select scouts R based on 10% to 20% of the number of sparrows. N The number of units is used to update the reconnaissance position using formula (14).

[0089]

[0090] in Let be the global optimal position for the sparrow population, β be the step size control parameter following a standard normal distribution, and K∈[-1, 1] be a random number. e is a small constant to avoid a denominator of 0. When At that time, scout sparrows are on the periphery of the group and are vulnerable to predators; when At that time, the scout sparrow is in the center of the population and needs to approach other sparrows to counter-predate.

[0091] (6) Use equation (10) to perform mutation operations.

[0092] (7) Output the optimal parameters when the maximum iteration N or the solution accuracy ε is satisfied. Otherwise, repeat steps 2 through 7 until the termination condition is met.

[0093] (8) Use the optimal parameters from step 7 to construct the ELM training and then output the results.

[0094] 2. LSSVR

[0095] This invention uses the aforementioned health indicators and volume sequences as input and output to train an LSSVR model. When the health indicators for a new cycle are extracted as input to the LSSVR model, the model can output the volume estimate corresponding to that cycle. The LSSVR regression model can be expressed as:

[0096]

[0097] Where N represents the total number of samples, γ i Let K(x) represent the weight vector. i x) is the kernel function, and b is the bias term.

[0098] Based on the above embodiments, the remaining lifespan of lithium batteries is predicted using the CSSA-ELM-LSSVR fusion algorithm, including:

[0099] Using a data fusion method, combining the advantages of ELM and LSSVR, and rationally allocating the weights of the predicted values ​​of ELM and LSSVR, the remaining lifespan of the lithium battery is predicted based on the CSSA-ELM-LSSVR fusion algorithm.

[0100] As can be seen from the above embodiments, the present invention combines the advantages of ELM and LSSVR to provide a new CSSA-ELM-LSSVR algorithm for RUL prediction. By weighting the predictions of the CSSA-ELM and LSSVR models, the prediction result given by the fusion model is obtained. The fusion model is expressed as:

[0101]

[0102] in and The values ​​are the prediction results of CSSA-ELM and LSSVR, respectively, where ω1 and ω2 are the corresponding weights, and ω1+ω2=1.

[0103] Understandably, the prediction accuracy of individual models varies; the larger the variance of the prediction error, the lower the prediction accuracy of a single model, and the lower its importance in the fusion model. The formula for calculating the aforementioned weights is as follows:

[0104]

[0105]

[0106] in and The covariance of the prediction error is respectively and It can be seen that the weights ω1 of the CSSA-ELM model and the prediction error covariance of the LSSVR model are related. It is directly proportional, and vice versa.

[0107] Based on the above embodiments, error analysis is performed on the prediction results, including:

[0108] Error analysis was performed on the prediction results. To verify the effectiveness of the model, the RUL prediction error E was used. rul Root mean square error (RMSE) and coefficient of determination (R²) 2 Evaluate the model performance.

[0109] As can be seen from the above embodiments, in order to verify the effectiveness of the prediction model, the embodiments of the present invention perform error analysis on the prediction results, and respectively use the RUL prediction error E rul Root mean square error (RMSE) and coefficient of determination (R²) 2 The model performance is evaluated using the following formula:

[0110] E rul =|RUL T -RUL P | (19)

[0111]

[0112]

[0113] Among them RUL T To predict the RUL of the starting point T, RUL P RUL is the predicted value of the fusion model. Q k Actual capacity This is the capacity forecast value. This represents the capacity average. N is the number of test sample periods. E rul The smaller the RMSE value, the better R 2 The larger the value, the better the RUL prediction performance.

[0114] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features, and such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should all be covered within the scope of the claims of the present invention.

Claims

1. A method for predicting the remaining lifespan of lithium batteries based on a fusion data-driven model, characterized in that, Includes the following steps: Obtain the capacity data of the lithium battery and plot the lithium battery capacity curve, as well as the constant current-constant voltage charging voltage curve and constant current discharging voltage curve for different cycle periods of the lithium battery; among them, the lithium battery capacity curve is the battery capacity-cycle period curve. The constant current charging duration, constant voltage charging duration, and vertical slope at the turning point of the constant current charging curve are extracted as health features based on the constant current-constant voltage charging voltage curve; the vertical slope at the turning point of the constant current discharging curve is extracted as a health feature based on the constant current discharging voltage curve. Using these four health features as input and the corresponding battery capacity as output, the lithium battery degradation process is trained on the training set based on the Chaotic Sparrow Extreme Learning Machine and the Least Squares Support Vector Regression Model, respectively, to obtain the trained Chaotic Sparrow Extreme Learning Machine Model and Least Squares Support Vector Regression Model. Two trained models are used to obtain battery capacity prediction values. The two battery capacity prediction values ​​are weighted to obtain the final lithium battery capacity prediction value. Finally, the remaining lifespan of the lithium battery is obtained by combining the lithium battery capacity curve. Specifically, two trained models are used to obtain predicted battery capacity values, and the two predicted battery capacity values ​​are weighted together to obtain the final predicted lithium battery capacity value. for: In the formula, and The figures represent the predicted lithium battery capacity values ​​from the Chaotic Sparrow-Extreme Learning Machine model and the Least Squares Support Vector Regression model, respectively. and The corresponding weights are calculated as follows: In the formula, and These are the covariances of the prediction errors of the Chaotic Sparrow Extreme Learning Machine and the Least Squares Support Vector Regression model on the training set, respectively.

2. The method for predicting the remaining lifespan of lithium batteries based on a fusion data-driven model according to claim 1, characterized in that, After extracting health features, the Pearson correlation coefficient was used to quantitatively analyze the correlation between health features and battery capacity, and health features with high correlation were selected as the input for model training.

3. The method for predicting the remaining lifespan of lithium batteries based on a fusion data-driven model according to claim 1, characterized in that, The lithium battery degradation process was trained using a chaotic sparrow-extreme learning machine and a least squares support vector regression model, respectively. For the Chaotic Sparrow Extreme Learning Machine, the initial weights of the Extreme Learning Machine are optimized using the Chaotic Sparrow algorithm. and bias The fitness function is designed according to the mean square error, and the obtained optimal parameters are used to construct and train the extreme learning machine to obtain the optimal chaotic sparrow-extreme learning machine model that reflects the degradation process of lithium battery. For the least squares support vector regression model, health indicators and battery capacity sequences are used as inputs and outputs to train the least squares support vector regression model in order to obtain the optimal least squares support vector regression model.

Images