Multi-level safety monitoring method and system for lithium ion battery system

Abstract

The invention discloses a multi-level safety monitoring method and system for a lithium ion battery energy storage system. The method comprises the following steps: calculating the spatial information entropy inconsistency of operation data of each battery cluster, judging whether the spatial information entropy inconsistency is too large or not, and triggering an alarm if the spatial information entropy inconsistency is too large; on the basis of judging that the spatial information entropy inconsistency of the operation data of the battery clusters is too large, calculating the spatial information entropy inconsistency of the operation data of each battery module, judging whether the spatial information entropy inconsistency is too large or not, and triggering an alarm if the spatial information entropy inconsistency is too large; according to the judgment result of the battery modules, selecting the battery modules exceeding the spatial information entropy inconsistency threshold value, and executing a time information entropy fault diagnosis algorithm on all the battery monomers in the battery modules; and if no battery module exceeds the spatial information entropy inconsistency threshold, selecting R battery modules by adopting a random sampling method to execute a time information entropy fault diagnosis algorithm. According to the method, the time information entropy and the correlation coefficient are combined, and early warning can be effectively carried out on gradual change faults.

Description

Technical Field

[0001] The present invention belongs to the technical field of lithium-ion batteries, and particularly relates to a multi-level safety monitoring method and system for a lithium-ion battery system based on information entropy. Background Art

[0002] In recent years, lithium-ion batteries have been widely used in new energy vehicles and energy storage systems due to their advantages such as high voltage, large specific capacity, low self-discharge rate, long cycle life, and high energy density. Lithium-ion batteries have become the most important power batteries for new energy vehicles at present. Lithium-ion battery energy storage is the second largest energy storage technology after pumped storage.

[0003] To meet certain voltage and capacity requirements, a power battery system or an energy storage system is usually composed of thousands of lithium-ion batteries connected in series and parallel, with complex battery structures and large inconsistencies among batteries. The inconsistencies among lithium-ion batteries will increase during charge and discharge, which will not only cause the decline of battery performance but also increase the risk of potential safety accidents. Therefore, it is necessary to conduct safety monitoring on the lithium-ion battery system to identify potential safety hazards at an early stage, so as to take appropriate measures in a timely manner to prevent the failure from further escalating into thermal runaway.

[0004] However, due to the complexity of the battery system, the existing safety monitoring methods still need to be improved in terms of calculation efficiency and effectiveness of fault detection. Summary of the Invention

[0005] Aiming at the deficiencies of the above-mentioned existing technologies, the present invention provides a multi-level safety monitoring method and system for a lithium-ion battery system based on information entropy, which combines time information entropy and correlation coefficient, can improve the calculation efficiency and can effectively give early warnings for gradual faults.

[0006] For this reason, the present invention adopts the following technical solutions.

[0007] In a first aspect, the present invention provides a multi-level safety monitoring method for a lithium-ion battery system, which includes: Calculating the spatial information entropy inconsistency of the operation data of each battery cluster, and determining whether it is too large. If it is too large, an alarm is triggered; On the basis of determining that the spatial information entropy inconsistency of the battery cluster operation data is too large, calculating the spatial information entropy inconsistency of the operation data of each battery module, and determining whether it is too large. If it is too large, an alarm is triggered; According to the judgment result of the battery module, select the battery modules that exceed the spatial information entropy inconsistency threshold, and perform a time information entropy fault diagnosis algorithm on all battery cells in the battery module; if no battery module exceeds the spatial information entropy inconsistency threshold, then use the random sampling method to select RA time information entropy fault diagnosis algorithm is executed for the battery module; The time information entropy fault diagnosis algorithm described above includes: Calculate the time information entropy of the battery cell operation data, and calculate the correlation coefficient of the time information entropy between the battery cells in the same battery module; Compare the average value of the correlation coefficient with a preset threshold to diagnose the faulty battery cell.

[0008] Furthermore, calculate the spatial information entropy inconsistency degree of the operation data of each battery cluster, and determine whether it is too large. If it is too large, trigger an alarm. Specifically, it includes: Calculate the first spatial information entropy of each battery cluster regarding the operation data at the q th acquisition moment; According to the first spatial information entropy, calculate the difference between it and the maximum spatial information entropy that satisfies the equiprobability distribution characteristic, as the spatial information entropy inconsistency degree index of the operation data of each battery cluster; Compare the spatial information entropy inconsistency degree index of the operation data of each battery cluster with the spatial information entropy inconsistency degree threshold of the battery cluster to determine whether the spatial information entropy inconsistency degree of the operation data of the battery cluster is too large. If it is too large, trigger an alarm.

[0009] Furthermore, the calculation formula of the first spatial information entropy is as follows: , X ={ ; n = 1, 2, …, N ; m = 1, 2, …, M ; k = 1, 2, …, K ; q = 1, 2, …, Q}, where, represents the first spatial information entropy, n is the serial number of the battery cell in the battery module, m is the serial number of the battery module in the battery cluster, k is the serial number of the battery cluster in the energy storage system, N is the number of battery cells contained in each battery module, M is the number of battery modules contained in each battery cluster, K is the number of battery clusters contained in the energy storage system, Q is the total number of samples; represents the th battery cluster, the th battery module in it, the n th battery cell in the battery module, and the qData at a collection moment; Represents the first probability; The aforementioned first probability Is calculated by the following formula: .

[0010] Furthermore, if all the lithium-ion batteries in the battery cluster have exactly the same values regarding the operation data X That is, the operation data X At this time, it satisfies the equiprobable distribution characteristic, then the first spatial information entropy Obtains the maximum value; The inconsistency index of the spatial information entropy of each battery cluster's operation data is calculated by the following formula: , In the formula, Is the first spatial information entropy The difference from the maximum information entropy that satisfies the equiprobable distribution characteristic.

[0011] Furthermore, if the inconsistency index of the spatial information entropy of each battery cluster's operation data is greater than or equal to the spatial information entropy inconsistency threshold of the battery cluster, it is determined that the inconsistency of the spatial information entropy of the battery cluster's operation data is too large.

[0012] Further, calculating the inconsistency index of the spatial information entropy of each battery module's operation data and determining whether it is too large. If it is too large, an alarm is triggered. Specifically, it includes: Calculating the second spatial information entropy of each battery module at the q th collection moment regarding the operation data; According to the second spatial information entropy, calculating the difference between it and the maximum spatial information entropy that satisfies the equiprobable distribution characteristic as the inconsistency index of the spatial information entropy of each battery module's operation data; Comparing the inconsistency index of the spatial information entropy of each battery module's operation data with the spatial information entropy inconsistency threshold of the battery module to determine whether the inconsistency of the spatial information entropy of the battery module's operation data is too large. If it is too large, an alarm is triggered.

[0013] Furthermore, the second spatial information entropy Is calculated by the following formula: , Wherein, Represents the second probability; The second probability Is calculated by the following formula: .

[0014] Furthermore, the inconsistency index of the spatial information entropy of each battery module's operation data is calculated by the following formula: , In the formula, is the second spatial information entropy and the difference between the maximum information entropy satisfying the equiprobability distribution characteristic.

[0015] Furthermore, the operating data includes temperature, current or voltage.

[0016] Furthermore, R The value is selected after comprehensively considering the calculation efficiency and system security requirements.

[0017] Furthermore, if the spatial information entropy inconsistency index of the operating data of each battery module is greater than or equal to the spatial information entropy inconsistency threshold of the battery module, it is determined that the spatial information entropy of the operating data of the battery module is too large.

[0018] In a second aspect, the present invention provides a multi-level safety monitoring system for a lithium-ion battery system, which includes: Battery cluster alarm trigger unit: used to calculate the spatial information entropy inconsistency of the operating data of each battery cluster, and determine whether it is too large. If it is too large, an alarm is triggered; Battery module alarm trigger unit: used to calculate the spatial information entropy inconsistency of the operating data of each battery module on the basis of determining that the spatial information entropy of the operating data of the battery cluster is too large, and determine whether it is too large. If it is too large, an alarm is triggered; Faulty battery cell diagnosis unit: according to the judgment result of the battery module, select the battery module exceeding the spatial information entropy inconsistency threshold, and execute the time information entropy fault diagnosis algorithm on all battery cells in the battery module; if no battery module exceeds the spatial information entropy inconsistency threshold, then use the random sampling method to select R battery modules to execute the time information entropy fault diagnosis algorithm.

[0019] Compared with the prior art, the beneficial effects of the present invention are as follows: First, the present invention can be applied to different hierarchical structures of power battery systems or energy storage systems; Second, through monitoring the spatial information entropy inconsistency of battery clusters (or battery packs) and battery modules, the fault is initially located. On this basis, battery modules with potential fault risks are selectively targeted to execute the time information entropy fault diagnosis algorithm at the battery cell level, thereby improving the calculation efficiency; Third, the present invention combines time information entropy with the correlation coefficient, which can effectively give early warning of gradual faults, so as to take appropriate measures in time to avoid the further escalation of the fault into thermal runaway. Description of the Drawings

[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention or in the prior art, the following will briefly introduce the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0021] Figure 1 is a flowchart of a multi-level safety monitoring method for a lithium-ion battery system of the present invention; Figure 2 is a specific implementation diagram of the multi-level safety monitoring method for the lithium-ion battery system in the specific implementation manner of the present invention; Figure 3 is a voltage diagram of all lithium-ion batteries in the sixth module of the first cluster of the energy storage system in the specific implementation manner of the present invention; Figure 4 is an information entropy diagram of all lithium-ion batteries in the sixth module of the first cluster of the energy storage system with respect to voltage in the specific implementation manner of the present invention; Figure 5 is an average correlation coefficient diagram of the information entropy of all lithium-ion batteries in the sixth module of the first cluster of the energy storage system with respect to voltage in the specific implementation manner of the present invention; Figure 6 is a composition diagram of a multi-level safety monitoring system for a lithium-ion battery system of the present invention. Specific implementation manner

[0022] In order to make the technical solutions of the present invention clearer, the present invention will be described in detail below in conjunction with the specification drawings and specific implementation manners.

[0023] Example 1 This embodiment is a multi-level safety monitoring method for a lithium-ion battery energy storage system based on information entropy. This method has two parts. The first part is an algorithm for monitoring the inconsistency degree of the spatial information entropy of battery clusters (or battery packs) and battery modules, and the second part is an algorithm for diagnosing time information entropy faults of battery cells. This case takes an energy storage system with a three-layer structure of "battery cluster - battery module - battery cell" as an example. This energy storage system altogether includes 10 independently parallel-connected battery clusters, and each battery cluster is composed of 38 series-connected battery modules. Each battery module altogether includes 6 lithium-ion battery cells.

[0024] See Figure 1 and Figure 2 , the specific steps of the above multi-level safety monitoring method for the lithium-ion battery energy storage system are as follows: Step 1: First, collect the latest operation data of different types, including temperature (T), current (I), voltage (U), etc. The collected operation data can be expressed as X ={ ; n = 1, 2, …, N ; m = 1, 2, …, M ; k = 1, 2, …, K ; q = 1, 2, …, Q}, where q is the acquisition time sequence number, n is the sequence number of the battery cell in the battery module, m is the sequence number of the battery module in the battery cluster, k is the sequence number of the battery cluster in the energy storage system, Q is the total number of samplings, N is the number of battery cells contained in each module, M is the number of battery modules contained in each battery cluster, K is the number of battery clusters contained in the energy storage system, X can represent data such as voltage, current, temperature, etc. In this embodiment, K is equal to 10, M is equal to 38, and N is equal to 6.

[0025] Step 2: Calculate the first spatial information entropy q of the X operation data at the -th acquisition time for each battery cluster; the first spatial information entropy is calculated by the following formula : (1) where the first probability is calculated by the following formula: (2) Equation (1) can evaluate the inconsistency degree of the spatial information entropy of the operation data at the q -th acquisition time among all the lithium-ion batteries in a battery cluster. If the values of the operation data X of all the lithium-ion batteries in this battery cluster are exactly the same, that is X and at this time it satisfies the equiprobability distribution characteristic, then reaches the maximum value.

[0026] Step 3, according to the first spatial information entropy of the operation data of each battery cluster obtained by calculation, calculate the difference between it and the maximum information entropy that satisfies the equiprobability distribution characteristic, as the inconsistency degree index, and calculate it by the following formula: (3) Step 4, the inconsistency degree index Compare with the battery cluster inconsistency threshold When comparing, when is greater than or equal to the pre-set battery cluster inconsistency threshold : (4) it is considered that the k battery cluster has excessive inconsistency at the q th acquisition moment, thus triggering an alarm. In this embodiment, ppm (10 -6 ) is selected as the unit.

[0027] Step 5, calculate the second spatial information entropy of each battery module at the q th acquisition moment with respect to the operation data X ; calculate through the following formula: (5) wherein, the second probability is calculated through the following formula: (6) Step 6, according to the second spatial information entropy of the operation data of each battery module calculated , calculate the difference between it and the maximum information entropy satisfying the equiprobability distribution characteristic (7) Step 7, compare the calculated inconsistency index of each battery module with the battery module inconsistency threshold When comparing, when is greater than or equal to the pre-set battery module inconsistency threshold , (8) it is considered that the k battery cluster's m battery module has excessive inconsistency at the q th acquisition moment, thus triggering an alarm.

[0028] Step 8, according to the judgment result of Step 7, select the battery module exceeding the inconsistency threshold, and execute the subsequent time information entropy fault diagnosis algorithm (i.e., Steps 9-12) for all battery cells in this battery module. If no battery module in a certain battery cluster exceeds the inconsistency threshold, use the random sampling method to select any 1 battery module from this battery cluster to execute the subsequent time information entropy fault diagnosis algorithm (i.e., Steps 9-12). As Figure 2 shown, at the qAt a sampling moment, for the case where no battery module exceeds the inconsistency threshold, the early warning cumulative count i is equal to 0. Therefore, the random sampling method is executed. By m i =int( rand ()× M ) + 1, randomly select the battery module numbered m i to execute the subsequent time information entropy fault diagnosis algorithm, where rand represents the function that generates random numbers between (0, 1).

[0029] Step 9: Extract a segment of operation data q of the battery cell up to the X th acquisition moment regarding the operation data , where n is the serial number of the battery cell in the k th cluster and the m th battery module, W is the size of the sliding window; Furthermore, reconstruct into , where D is the size of the reconstruction vector , is in , min( ) + the frequency; then the information entropy k of the m th battery module in the n th cluster and the q th battery cell at the X th time step regarding the data can be calculated by the following formula, (9) where is the probability, calculated by the following formula: (10) For example, assume that through Steps 1 - 8, it has been detected that the sixth module in the first cluster is abnormal. Then in Step 9, execute the "time" information entropy fault diagnosis algorithm on all lithium - ion batteries in this module. Figure 3 shows the voltage change curves of all battery cells in the sixth battery module of the first battery cluster of the energy storage system on a certain day. It can be seen that due to the micro - internal short - circuit state of the second battery, its voltage is only slightly lower than that of normal batteries, and the difference between the voltage curves of different batteries is very small. The faulty cell cannot be detected directly through the voltage curves. Figure 4The "time" information entropy change curve of all battery cells of the sixth battery module in the first battery cluster on a certain day with respect to voltage is given. It can be seen that the information entropy differences of all normal lithium-ion battery cells with respect to voltage are very small, while the relative deviation between the second battery and the normal batteries in this module can reach up to 7%. The information entropy effectively amplifies the differences between faulty cells and normal cells. According to the information entropy analysis results, it can be detected that the second battery may be abnormal.

[0030] Step 10: Calculate the correlation coefficient of information entropy between different battery cells in the same module , where n and are two battery cells belonging to the k th cluster and the m th module, and are calculated by the following formula: (11) Step 11: For the convenience of comparison, further calculate the average value of the correlation coefficients of information entropy between each battery and other batteries: (12) represents the information entropy of the k th battery cell in the m th module of the n th cluster at the q th time step, and and the information entropy of other batteries in the same module at the q th time step ( ).

[0031] Step 12: Finally, compare the average value of the correlation coefficients of information entropy between each battery and other batteries calculated with the preset threshold . If (13) , it indicates that the k th battery cell in the m th module of the n th cluster has a potential fault risk and triggers an early warning.

[0032] As Figure 5 shown, calculate the average correlation coefficient of the voltage information entropy of all cells in the sixth module of the first cluster It can be seen that the average correlation coefficient of the information entropy of the second battery in the sixth module of the first cluster with respect to voltage is significantly lower than that of other battery cells in the same module with respect to voltage. Moreover, the minimum value of the average correlation coefficient of the information entropy of the second battery is less than the alarm threshold (0.7) and remains below the threshold for a certain period of time. Therefore, the second battery in the micro internal short - circuit state can be effectively detected through the average correlation coefficient of the information entropy.

[0033] Embodiment 2 This embodiment is a multi - level safety monitoring system for a lithium - ion battery system. As Figure 6 shown, it consists of a battery cluster alarm trigger unit, a battery module alarm trigger unit, and a faulty battery cell diagnosis unit.

[0034] The described battery cluster alarm trigger unit: is used to calculate the spatial information entropy inconsistency degree of the operating data of each battery cluster, and determine whether it is too large. If it is too large, an alarm is triggered. The described battery module alarm trigger unit: is used to calculate the spatial information entropy inconsistency degree of the operating data of each battery module on the basis of determining that the spatial information entropy inconsistency degree of the battery cluster operating data is too large, and determine whether it is too large. If it is too large, an alarm is triggered. The described faulty battery cell diagnosis unit: according to the judgment result of the battery module, selects the battery module that exceeds the spatial information entropy inconsistency threshold, and executes the time - information entropy fault diagnosis algorithm on all battery cells in this battery module; if no battery module exceeds the spatial information entropy inconsistency threshold, the random sampling method is used to select R battery modules to execute the time - information entropy fault diagnosis algorithm.

[0035] It should be noted that each unit in the above - mentioned multi - level safety monitoring system for a lithium - ion battery system can be implemented in whole or in part through software, hardware, and their combination. The above - mentioned units can be embedded in the processor of the computer device in hardware form or be independent of it, or can be stored in the memory of the computer device in software form, so that the processor can call and execute the operations corresponding to the above - mentioned units. For the specific limitations of the multi - level safety monitoring system for a lithium - ion battery system, refer to the limitations of the multi - level safety monitoring method for a lithium - ion battery system (i.e., Embodiment 1) in the above text. The two have the same functions and effects, and will not be elaborated here.

[0036] The above embodiments are only used to illustrate the technical concept and characteristics of the present invention. The purpose is to enable those of ordinary skill in the art to understand the content of the present invention and implement it accordingly, and cannot be used to limit the protection scope of the present invention. Any equivalent changes or modifications made according to the essence of the content of the present invention should be covered within the protection scope of the present invention.

Claims

1. A multi-level safety monitoring method for a lithium-ion battery system, characterized in that Including: Calculating the spatial information entropy inconsistency degree of the operation data of each battery cluster, and determining whether it is too large. If it is too large, an alarm is triggered; On the basis of determining that the spatial information entropy inconsistency degree of the operation data of the battery cluster is too large, calculating the spatial information entropy inconsistency degree of the operation data of each battery module, and determining whether it is too large. If it is too large, an alarm is triggered; According to the judgment result of the battery module, selecting the battery modules that exceed the spatial information entropy inconsistency degree threshold, and performing a time information entropy fault diagnosis algorithm on all battery cells in the battery module; If none of the battery modules exceeds the spatial information entropy inconsistency threshold, the random sampling method is used to select R battery modules to execute the time information entropy fault diagnosis algorithm; The time information entropy fault diagnosis algorithm described above includes: Calculating the time information entropy of the operation data of the battery cell, and calculating the correlation coefficient of the time information entropy between the battery cells in the same battery module; Comparing the average value of the correlation coefficient with a preset threshold to diagnose the faulty battery cell.

2. The multi-level safety monitoring method for a lithium-ion battery system according to claim 1, wherein The calculation of the spatial information entropy inconsistency degree of the operation data of each battery cluster, and determining whether it is too large. If it is too large, an alarm is triggered, specifically including: Calculate the first spatial information entropy of the operating data at the q th acquisition moment for each battery cluster; Calculating the difference between the first spatial information entropy and the maximum spatial information entropy that satisfies the equiprobability distribution characteristic as the spatial information entropy inconsistency degree index of the operation data of each battery cluster; Comparing the spatial information entropy inconsistency degree index of the operation data of each battery cluster with the spatial information entropy inconsistency degree threshold of the battery cluster to determine whether the spatial information entropy inconsistency degree of the operation data of the battery cluster is too large. If it is too large, an alarm is triggered.

3. A multi-level safety monitoring method for a lithium-ion battery system according to claim 2, characterized in that The calculation formula of the first spatial information entropy is as follows: ， X ={ ; n =1,2, …, N ; m =1,2,…, M ; k =1,2,…, K ; q =1,2,…, Q}， Among them, represents the first spatial information entropy, n is the serial number of the battery cell in the battery module, m is the serial number of the battery module in the battery cluster, k is the serial number of the battery cluster in the energy storage system, N is the number of battery cells contained in each battery module, M is the number of battery modules contained in each battery cluster, K is the number of battery clusters contained in the energy storage system, Q is the total number of samplings; represents the th th battery module in the n th battery cell in the q th acquisition moment; represents the first probability; The first probability described above is calculated by the following formula: 。 4. A multi-level safety monitoring method for a lithium-ion battery system according to claim 3, characterized in that, If all the lithium-ion batteries in the battery cluster have exactly the same values for the operating data X i.e., the operating data X satisfies the equiprobable distribution characteristic at this time, then the first spatial information entropy reaches the maximum value; the inconsistency index of the spatial information entropy of the operating data of each battery cluster is calculated by the following formula: ， In the formula, is the difference between the first spatial information entropy and the maximum information entropy that satisfies the equiprobability distribution characteristic.

5. A multi-level safety monitoring method for a lithium-ion battery system according to claim 3, characterized in that If the spatial information entropy inconsistency degree index of the operation data of each battery cluster is greater than or equal to the spatial information entropy inconsistency degree threshold of the battery cluster, it is determined that the spatial information entropy of the operation data of the battery cluster is too large.

6. A multi-level safety monitoring method for a lithium-ion battery system according to claim 1, characterized in that The calculation of the spatial information entropy inconsistency degree of the operation data of each battery module, and determining whether it is too large. If it is too large, an alarm is triggered, specifically including: Calculate the second spatial information entropy of the operating data at the q th acquisition moment for each battery module; Calculating the difference between the second spatial information entropy and the maximum spatial information entropy that satisfies the equiprobability distribution characteristic as the spatial information entropy inconsistency degree index of the operation data of each battery module; Comparing the spatial information entropy inconsistency degree index of the operation data of each battery module with the spatial information entropy inconsistency degree threshold of the battery module to determine whether the spatial information entropy inconsistency degree of the operation data of the battery module is too large. If it is too large, an alarm is triggered.

7. A multi-level safety monitoring method for a lithium-ion battery system according to claim 6, characterized in that, The second spatial information entropy is calculated by the following formula: ， Among them, represents the second probability; the second probability is calculated by the following formula: 。 8. A multi-level safety monitoring method for a lithium-ion battery system according to claim 6, characterized in that, The spatial information entropy inconsistency degree index of the operation data of each battery module is calculated by the following formula: ， Wherein, is the second spatial information entropy and the difference between it and the maximum information entropy that satisfies the equiprobability distribution characteristic.

9. A multi-level safety monitoring method for a lithium-ion battery system according to claim 6, characterized in that If the spatial information entropy inconsistency degree index of the operation data of each battery module is greater than or equal to the spatial information entropy inconsistency degree threshold of the battery module, it is determined that the spatial information entropy of the operation data of the battery module is too large.

10. A multi-level safety monitoring system for a lithium-ion battery system, which is used to implement the multi-level safety monitoring method for the lithium-ion battery system according to any one of claims 1-9, characterized in that, Including: Battery cluster alarm trigger unit: used to calculate the spatial information entropy inconsistency degree of the operation data of each battery cluster, and determine whether it is too large. If it is too large, an alarm is triggered; Battery module alarm trigger unit: used to calculate the spatial information entropy inconsistency degree of the operation data of each battery module on the basis of determining that the spatial information entropy inconsistency degree of the operation data of the battery cluster is too large, and determine whether it is too large. If it is too large, an alarm is triggered; Faulty battery cell diagnosis unit: According to the judgment result of the battery module, select the battery module whose spatial information entropy inconsistency degree exceeds the threshold, and execute the time information entropy fault diagnosis algorithm on all battery cells in the battery module; If none of the battery modules exceeds the spatial information entropy inconsistency threshold, the random sampling method is used to select R battery modules to execute the time information entropy fault diagnosis algorithm.

Images